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Mathematics LibreTexts

11.E: Parametric Equations and Polar Coordinates (Exercises)

  • Page ID
    3606
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    11.1: Parametric Equations

    For the following exercises, sketch the curves below by eliminating the parameter t. Give the orientation of the curve.

    1) \(\displaystyle x=t^2+2t, y=t+1\)

    Solution: orientation: bottom to top

    A parabola open to the right with (−1, 0) being the point furthest the left with arrow going from the bottom through (−1, 0) and up.

    2) \(\displaystyle x=cos(t),y=sin(t),(0,2π]\)

    3) \(\displaystyle x=2t+4,y=t−1\)

    Solution: orientation: left to right

    A straight line passing through (0, −3) and (6, 0) with arrow pointing up and to the right.

    4) \(\displaystyle x=3−t,y=2t−3,1.5≤t≤3\)

    For the following exercises, eliminate the parameter and sketch the graphs.

    5) \(\displaystyle x=2t^2,y=t^4+1\)

    Solution: \(\displaystyle y=\frac{x^2}{4}+1\)

    Half a parabola starting at the origin and passing through (2, 2) with arrow pointed up and to the right.

    For the following exercises, use technology (CAS or calculator) to sketch the parametric equations.

    6) [T] \(\displaystyle x=t^2+t,y=t^2−1\)

    7) [T] \(\displaystyle x=e^{−t},y=e^{2t}−1\)

    Solution:

    A curve going through (1, 0) and (0, 3) with arrow pointing up and to the left.

    8) [T] \(\displaystyle x=3cost,y=4sint\)

    9) [T] \(\displaystyle x=sect,y=cost\)

    A graph with asymptotes at the x and y axes. There is a portion of the graph in the third quadrant with arrow pointing down and to the right. There is a portion of the graph in the first quadrant with arrow pointing down and to the right.

    For the following exercises, sketch the parametric equations by eliminating the parameter. Indicate any asymptotes of the graph.

    10) \(\displaystyle x=e^t,y=e^{2t}+1\)

    11) \(\displaystyle x=6sin(2θ),y=4cos(2θ)\)

    Solution:

    An ellipse with minor axis vertical and of length 8 and major axis horizontal and of length 12 that is centered at the origin. The arrows go counterclockwise.

    12) \(\displaystyle x=cosθ,y=2sin(2θ)\)

    13) \(\displaystyle x=3−2cosθ,y=−5+3sinθ\)

    Solution:

    An ellipse in the fourth quadrant with minor axis horizontal and of length 4 and major axis vertical and of length 6. The arrows go clockwise.

    14) \(\displaystyle x=4+2cosθ,y=−1+sinθ\)

    15) \(\displaystyle x=sect,y=tant\)

    Solution: Asymptotes are \(\displaystyle y=x\) and \(\displaystyle y=−x\)

    A graph with asymptotes at y = x and y = −x. The first part of the graph occurs in the second and third quadrants with vertex at (−1, 0). The second part of the graph occurs in the first and fourth quadrants with vertex as (1, 0).

    16) \(\displaystyle x=ln(2t),y=t^2\)

    17) \(\displaystyle x=e^t,y=e^{2t}\)

    Solution:

    A curve starting slightly above the origin and increasing to the right with arrow pointing up and to the right.

    18) \(\displaystyle x=e^{−2t},y=e^{3t}\)

    19) \(\displaystyle x=t^3,y=3lnt\)

    Solution:

    A curve with asymptote being the y axis. The curve starts in the fourth quadrant and increases rapidly through (1, 0) at which point is increases much more slowly.

    20) \(\displaystyle x=4secθ,y=3tanθ\)

    For the following exercises, convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form.

    21) \(\displaystyle x=t^2−1,y=\frac{t}{2}\)

    Solution: \(\displaystyle x=4y^2−1;\) domain: \(\displaystyle x∈[1,∞)\).

    22) \(\displaystyle x=\frac{1}{\sqrt{t+1}},y=\frac{t}{1+t},t>−1\)

    23) \(\displaystyle x=4cosθ,y=3sinθ,t∈(0,2π]\)

    Solution: \(\displaystyle \frac{x^2}{16}+\frac{y^2}{9}=1;\) domain \(\displaystyle x∈[−4,4].\)

    24) \(\displaystyle x=cosht,y=sinht\)

    25) \(\displaystyle x=2t−3,y=6t−7\)

    Solution: \(\displaystyle y=3x+2;\) domain: all real numbers.

    26) \(\displaystyle x=t^2,y=t^3\)

    27) \(\displaystyle x=1+cost,y=3−sint\)

    Solution: \(\displaystyle (x−1)^2+(y−3)^2=1\); domain: \(\displaystyle x∈[0,2]\).

    28) \(\displaystyle x=\sqrt{t},y=2t+4\)

    29) \(\displaystyle x=sect,y=tant,π≤t<\frac{3π}{2}\)

    Solution: \(\displaystyle y=\sqrt{x^2−1}\); domain: \(\displaystyle x∈[−1,1]\).

    30) \(\displaystyle x=2cosht,y=4sinht\)

    31) \(\displaystyle x=cos(2t),y=sint\)

    Solution: \(\displaystyle y^2=\frac{1−x}{2};\) domain: \(\displaystyle x∈[2,∞)∪(−∞,−2].\)

    32) \(\displaystyle x=4t+3,y=16t^2−9\)

    33) \(\displaystyle x=t^2,y=2lnt,t≥1\)

    Solution: \(\displaystyle y=lnx;\) domain: \(\displaystyle x∈(0,∞).\)

    34) \(\displaystyle x=t3^,y=3lnt,t≥1\)

    35) \(\displaystyle x=t^n,y=nlnt,t≥1,\) where n is a natural number

    Solution: \(\displaystyle y=lnx;\) domain: \(\displaystyle x∈(0,∞).\)

    36) \(\displaystyle x=ln(5t)\) \(\displaystyle y=ln(t^2)\) where \(\displaystyle 1≤t≤e\)

    37) \(\displaystyle x=2sin(8t)\) \(\displaystyle y=2cos(8t)\)

    Solution: \(\displaystyle x^2+y^2=4;\) domain: \(\displaystyle x∈[−2,2].\)

    38) \(\displaystyle x=tant\) \(\displaystyle y=sec^2t−1\)

    For the following exercises, the pairs of parametric equations represent lines, parabolas, circles, ellipses, or hyperbolas. Name the type of basic curve that each pair of equations represents.

    39) \(\displaystyle x=3t+4\) \(\displaystyle y=5t−2\)

    Solutin: line

    40) \(\displaystyle x−4=5t\) \(\displaystyle y+2=t\)

    41) \(\displaystyle x=2t+1\) \(\displaystyle y=t^2−3\)

    Solution: parabola

    42) \(\displaystyle x=3cost\) \(\displaystyle y=3sint\)

    43) \(\displaystyle x=2cos(3t)\) \(\displaystyle y=2sin(3t)\)

    Solution: circle

    44) \(\displaystyle x=cosht\) \(\displaystyle y=sinht\)

    45) \(\displaystyle x=3cost\) \(\displaystyle y=4sint\)

    Solutin: ellipse

    46) \(\displaystyle x=2cos(3t)\) \(\displaystyle y=5sin(3t)\)

    47) \(\displaystyle x=3cosh(4t)\) \(\displaystyle y=4sinh(4t)\)

    Solution: hyperbola

    48) \(\displaystyle x=2cosht\) \(\displaystyle y=2sinht\)

    49) Show that \(\displaystyle x=h+rcosθ\) \(\displaystyle y=k+rsinθ\) represents the equation of a circle.

    50) Use the equations in the preceding problem to find a set of parametric equations for a circle whose radius is 5 and whose center is \(\displaystyle (−2,3)\).

    For the following exercises, use a graphing utility to graph the curve represented by the parametric equations and identify the curve from its equation.

    51) [T] \(\displaystyle x=θ+sinθ\) \(\displaystyle y=1−cosθ\)

    Solution: The equations represent a cycloid.

    A graph starting at (−6, 0) increasing rapidly to a sharp point at (−3, 2) and then decreasing rapidly to the origin. The graph is symmetric about the y axis, so the graph increases rapidly to (3, 2) before decreasing rapidly to (6, 0).

    52) [T] \(\displaystyle x=2t−2sint\) \(\displaystyle y=2−2cost\)

    53) [T] \(\displaystyle x=t−0.5sint\) \(\displaystyle y=1−1.5cost\)

    Solution:

    A graph starting at roughly (−6, 0) increasing to a rounded point and then decreasing to roughly (0, −0.5). The graph is symmetric about the y axis, so the graph increases to a rounded point before decreasing to roughly (6, 0).

    54) An airplane traveling horizontally at 100 m/s over flat ground at an elevation of 4000 meters must drop an emergency package on a target on the ground. The trajectory of the package is given by \(\displaystyle x=100t,y=−4.9t^2+4000,t≥0\) where the origin is the point on the ground directly beneath the plane at the moment of release. How many horizontal meters before the target should the package be released in order to hit the target?

    55) The trajectory of a bullet is given by \(\displaystyle x=v_0(cosα)ty=v_0(sinα)t−\frac{1}{2}gt^2\) where \(\displaystyle v_0=500m/s, g=9.8=9.8m/s^2\), and \(\displaystyle α=30degrees.\) When will the bullet hit the ground? How far from the gun will the bullet hit the ground?

    Solution: 22,092 meters at approximately 51 seconds.

    56) [T] Use technology to sketch the curve represented by \(\displaystyle x=sin(4t),y=sin(3t),0≤t≤2π\).

    57) [T] Use technology to sketch \(\displaystyle x=2tan(t),y=3sec(t),−π<t<π.\)

    Solution:

    A graph with asymptotes roughly near y = x and y = −x. The first part of the graph is in the first and second quadrants with vertex near (0, 3). The second part of the graph is in the third and fourth quadrants with vertex near (0, −3).

    58) Sketch the curve known as an epitrochoid, which gives the path of a point on a circle of radius b as it rolls on the outside of a circle of radius a. The equations are

    \(\displaystyle x=(a+b)cost−c⋅cos[\frac{(a+b)t}{b}]\) \(\displaystyle y=(a+b)sint−c⋅sin[\frac{(a+b)t}{b}]\).

    Let \(\displaystyle a=1,b=2,c=1.

    59) [T] Use technology to sketch the spiral curve given by \(\displaystyle x=tcos(t),y=tsin(t)\) from \(\displaystyle −2π≤t≤2π.\)

    A graph starting at roughly (−6, −1) decreasing to a minimum in the third quadrant near (−1, −4.8) increasing through roughly (0, −4.7) and (3, 0) to a maximum near (1, 1.9) before decreasing through (0, 1.5) to the origin. The graph is symmetric about the y axis, so the graph increases through (0, 1.5) to a maximum in the second quadrant, decreases again through (0, −4.7), and then increases to (6, −1).

    60) [T] Use technology to graph the curve given by the parametric equations \(\displaystyle x=2cot(t),y=1−cos(2t),−π/2≤t≤π/2.\) This curve is known as the witch of Agnesi.

    61) [T] Sketch the curve given by parametric equations \(\displaystyle x=cosh(t)\) \(\displaystyle y=sinh(t),\) where \(\displaystyle −2≤t≤2.\)

    Solution:

    A vaguely parabolic graph with vertex at the origin that is open to the right.

    11.2: Calculus of Parametric Curves

    For the following exercises, each set of parametric equations represents a line. Without eliminating the parameter, find the slope of each line.

    1) \(\displaystyle x=3+t,y=1−t\)

    2) \(\displaystyle x=8+2t,y=1\)

    Solution: 0

    3) \(\displaystyle x=4−3t,y=−2+6t\)

    4) \(\displaystyle x=−5t+7,y=3t−1\)

    Solution: \(\displaystyle \frac{−3}{5}\)

    For the following exercises, determine the slope of the tangent line, then find the equation of the tangent line at the given value of the parameter.

    5) \(\displaystyle x=3sint,y=3cost,t=\frac{π}{4}\)

    6) \(\displaystyle x=cost,y=8sint,t=\frac{π}{2}\)

    Solution: \(\displaystyle Slope=0; y=8.\)

    7) \(\displaystyle x=2t,y=t^3,t=−1\)

    8) \(\displaystyle x=t+\frac{1}{t},y=t−\frac{1}{t},t=1\)

    Solution: Slope is undefined; \(\displaystyle x=2\).

    9) \(\displaystyle x=\sqrt{t},y=2t,t=4\)

    For the following exercises, find all points on the curve that have the given slope.

    10) \(\displaystyle x=4cost,y=4sint,\) slope = 0.5

    Solution: \(\displaystyle t=arctan(−2); (\frac{4}{\sqrt{5}},\frac{−8}{\sqrt{5}})\).

    11) \(\displaystyle x=2cost,y=8sint,slope=−1\)

    12) \(\displaystyle x=t+\frac{1}{t},y=t−\frac{1}{t},slope=1\)

    Solution: No points possible; undefined expression.

    13) \(\displaystyle x=2+\sqrt{t},y=2−4t,slope=0\)

    For the following exercises, write the equation of the tangent line in Cartesian coordinates for the given parameter t.

    14) \(\displaystyle x=e^{\sqrt{t}},y=1−lnt^2,t=1\)

    Solution: \(\displaystyle y=−(\frac{2}{e})x+3\)

    15) \(\displaystyle x=tlnt,y=sin^2t,t=\frac{π}{4}\)

    16) \(\displaystyle x=e^t,y=(t−1)^2,at(1,1)\)

    Solution: \(\displaystyle y=2x−7\)

    17) For \(\displaystyle x=sin(2t),y=2sint\) where \(\displaystyle 0≤t<2π.\) Find all values of t at which a horizontal tangent line exists.

    18) For \(\displaystyle x=sin(2t),y=2sint\) where \(\displaystyle 0≤t<2π\). Find all values of t at which a vertical tangent line exists.

    Solution: \(\displaystyle \frac{π}{4},\frac{5π}{4},\frac{3π}{4},\frac{7π}{4}\)

    19) Find all points on the curve \(\displaystyle x=4cos(t),y=4sin(t)\) that have the slope of \(\displaystyle \frac{1}{2}\).

    20) Find \(\displaystyle \frac{dy}{dx}\) for \(\displaystyle x=sin(t),y=cos(t)\).

    Solution: \(\displaystyle \frac{dy}{dx}=−tan(t)\)

    21) Find the equation of the tangent line to \(\displaystyle x=sin(t),y=cos(t)\) at \(\displaystyle t=\frac{π}{4}\).

    22) For the curve \(\displaystyle x=4t,y=3t−2,\) find the slope and concavity of the curve at \(\displaystyle t=3\).

    Solution: \(\displaystyle \frac{dy}{dx}=\frac{3}{4}\) and \(\displaystyle \frac{d^2y}{dx^2}=0\), so the curve is neither concave up nor concave down at \(\displaystyle t=3\). Therefore the graph is linear and has a constant slope but no concavity.

    23) For the parametric curve whose equation is \(\displaystyle x=4cosθ,y=4sinθ\), find the slope and concavity of the curve at \(\displaystyle θ=\frac{π}{4}\).

    24) Find the slope and concavity for the curve whose equation is \(\displaystyle x=2+secθ,y=1+2tanθ\) at \(\displaystyle θ=\frac{π}{6}\).

    Solution: \(\displaystyle \frac{dy}{dx}=4,\frac{d^2y}{dx^2}=−6\sqrt{3};\) the curve is concave down at \(\displaystyle θ=\frac{π}{6}\).

    25) Find all points on the curve \(\displaystyle x=t+4,y=t^3−3t\) at which there are vertical and horizontal tangents.

    26) Find all points on the curve \(\displaystyle x=secθ,y=tanθ\) at which horizontal and vertical tangents exist.

    Solution: No horizontal tangents. Vertical tangents at \(\displaystyle (1,0),(−1,0)\).

    For the following exercises, find \(\displaystyle d^2y/dx^2\).

    27) \(\displaystyle x=t^4−1,y=t−t^2\)

    28) \(\displaystyle x=sin(πt),y=cos(πt)\)

    Solution: \(\displaystyle −sec^3(πt)\)

    29) \(\displaystyle x=e^{−t},y=te^{2t}\)

    For the following exercises, find points on the curve at which tangent line is horizontal or vertical.

    30) \(\displaystyle x=t(t^2−3),y=3(t^2−3)\)

    Solution: Horizontal \(\displaystyle (0,−9)\); vertical \(\displaystyle (±2,−6).\)

    31) \(\displaystyle x=\frac{3t}{1+t^3},y=\frac{3t^2}{1+t^3}\)

    For the following exercises, find \(\displaystyle dy/dx\) at the value of the parameter.

    32) \(\displaystyle x=cost,y=sint,t=\frac{3π}{4}\)

    Solution: 1

    33) \(\displaystyle x=\sqrt{t},y=2t+4,t=9\)

    34) \(\displaystyle x=4cos(2πs),y=3sin(2πs),s=−\frac{1}{4}\)

    Solution: 0

    For the following exercises, find \(\displaystyle d^2y/dx^2\) at the given point without eliminating the parameter.

    35) \(\displaystyle x=\frac{1}{2}t^2,y=\frac{1}{3}t^3,t=2\)

    36) \(\displaystyle x=\sqrt{t},y=2t+4,t=1\)

    Solution: 4

    37) Find t intervals on which the curve \(\displaystyle x=3t^2,y=t^3−t\) is concave up as well as concave down.

    38) Determine the concavity of the curve \(\displaystyle x=2t+lnt,y=2t−lnt\).

    Solution: Concave up on \(\displaystyle t>0\).

    39) Sketch and find the area under one arch of the cycloid \(\displaystyle x=r(θ−sinθ),y=r(1−cosθ)\).

    40) Find the area bounded by the curve \(\displaystyle x=cost,y=e^t,0≤t≤\frac{π}{2}\) and the lines \(\displaystyle y=1\) and \(\displaystyle x=0\).

    Solution: 1

    41) Find the area enclosed by the ellipse \(\displaystyle x=acosθ,y=bsinθ,0≤θ<2π.\)

    42) Find the area of the region bounded by \(\displaystyle x=2sin^2θ,y=2sin^2θtanθ\), for \(\displaystyle 0≤θ≤\frac{π}{2}\).

    Solution: \(\displaystyle \frac{3π}{2}\)

    For the following exercises, find the area of the regions bounded by the parametric curves and the indicated values of the parameter.

    43) \(\displaystyle x=2cotθ,y=2sin^2θ,0≤θ≤π\)

    44) [T] \(\displaystyle x=2acost−acos(2t),y=2asint−asin(2t),0≤t<2π\)

    Solution: \(\displaystyle 6πa^2\)

    45) [T] \(\displaystyle x=asin(2t),y=bsin(t),0≤t<2π\) (the “hourglass”)

    46) [T] \(\displaystyle x=2acost−asin(2t),y=bsint,0≤t<2π\) (the “teardrop”)

    Solution: \(\displaystyle 2πab\)

    For the following exercises, find the arc length of the curve on the indicated interval of the parameter.

    47) \(\displaystyle x=4t+3,y=3t−2,0≤t≤2\)

    48) \(\displaystyle x=\frac{1}{3}t^3,y=\frac{1}{2}t^2,0≤t≤1\)

    Soluton: \(\displaystyle \frac{1}{3}(2\sqrt{2}−1)\)

    49) \(\displaystyle x=cos(2t),y=sin(2t),0≤t≤\frac{π}{2}\)

    50) \(\displaystyle x=1+t^2,y=(1+t)^3,0≤t≤1\)

    Solution: 7.075

    51) \(\displaystyle x=e^tcost,y=e^tsint,0≤t≤\frac{π}{2}\) (express answer as a decimal rounded to three places)

    52) \(\displaystyle x=acos^3θ,y=asin^3θ\) on the interval \(\displaystyle [0,2π)\) (the hypocycloid)

    Solution: \(\displaystyle 6a\)

    53) Find the length of one arch of the cycloid \(\displaystyle x=4(t−sint),y=4(1−cost).\)

    54) Find the distance traveled by a particle with position \(\displaystyle (x,y)\) as t varies in the given time interval: \(\displaystyle x=sin^2t,y=cos^2t,0≤t≤3π\).

    Solution: \(\displaystyle 6\sqrt{2}\)

    55) Find the length of one arch of the cycloid \(\displaystyle x=θ−sinθ,y=1−cosθ\).

    56) Show that the total length of the ellipse \(\displaystyle x=4sinθ,y=3cosθ\) is \(\displaystyle L=16∫^{π/2}_0\sqrt{1−e^2sin^2θ}dθ\), where \(\displaystyle e=\frac{c}{a}\) and \(\displaystyle c=\sqrt{a^2−b^2}\).

    57) Find the length of the curve \(\displaystyle x=e^t−t,y=4e^{t/2},−8≤t≤3.\)

    For the following exercises, find the area of the surface obtained by rotating the given curve about the x-axis.

    58) \(\displaystyle x=t^3,y=t^2,0≤t≤1\)

    Solution: \(\displaystyle \frac{2π(247\sqrt{13}+64)}{1215}\)

    59) \(\displaystyle x=acos^3θ,y=asin^3θ,0≤θ≤\frac{π}{2}\)

    60) [T] Use a CAS to find the area of the surface generated by rotating \(\displaystyle x=t+t^3,y=t−\frac{1}{t^2},1≤t≤2\) about the x-axis. (Answer to three decimal places.)

    Solution: 59.101

    61) Find the surface area obtained by rotating \(\displaystyle x=3t^2,y=2t^3,0≤t≤5\) about the y-axis.

    62) Find the area of the surface generated by revolving \(\displaystyle x=t^2,y=2t,0≤t≤4\) about the x-axis.

    Solution: \(\displaystyle \frac{8π}{3}(17\sqrt{17}−1)\)

    63) Find the surface area generated by revolving \(\displaystyle x=t^2,y=2t^2,0≤t≤1\) about the y-axis.

    11.3: Polar Coordinates

    In the following exercises, plot the point whose polar coordinates are given by first constructing the angle \(\displaystyle θ\) and then marking off the distance r along the ray.

    1) \(\displaystyle (3,\frac{π}{6})\)

    Solution:

    On the polar coordinate plane, a ray is drawn from the origin marking π/6 and a point is drawn when this line crosses the circle with radius 3.

    2) \(\displaystyle (−2,\frac{5π}{3})\)

    3) \(\displaystyle (0,\frac{7π}{6})\)

    Solution:

    On the polar coordinate plane, a ray is drawn from the origin marking 7π/6 and a point is drawn when this line crosses the circle with radius 0, that is, it marks the origin.

    4) \(\displaystyle (−4,\frac{3π}{4})\)

    5) \(\displaystyle (1,\frac{π}{4})\)

    Solution:

    On the polar coordinate plane, a ray is drawn from the origin marking π/4 and a point is drawn when this line crosses the circle with radius 1.

    6) \(\displaystyle (2,\frac{5π}{6})\)

    7) \(\displaystyle (1,\frac{π}{2})\)

    Solution:

    On the polar coordinate plane, a ray is drawn from the origin marking π/2 and a point is drawn when this line crosses the circle with radius 1.

    For the following exercises, consider the polar graph below. Give two sets of polar coordinates for each point.

    The polar coordinate plane is divided into 12 pies. Point A is drawn on the first circle on the first spoke above the θ = 0 line in the first quadrant. Point B is drawn in the fourth quadrant on the third circle and the second spoke below the θ = 0 line. Point C is drawn on the θ = π line on the third circle. Point D is drawn on the fourth circle on the first spoke below the θ = π line.

    8) Coordinates of point A.

    9) Coordinates of point B.

    Solution: \(\displaystyle B(3,\frac{−π}{3}) B(−3,\frac{2π}{3})\)

    10) Coordinates of point C.

    11) Coordinates of point D.

    Solution: \(\displaystyle D(5,\frac{7π}{6}) D(−5,\frac{π}{6})\)

    For the following exercises, the rectangular coordinates of a point are given. Find two sets of polar coordinates for the point in \(\displaystyle (0,2π]\). Round to three decimal places.

    12) \(\displaystyle (2,2)\)

    13) \(\displaystyle (3,−4) (3, −4)\)

    Solution: \(\displaystyle (5,−0.927)(−5,−0.927+π)\)

    14) \(\displaystyle (8,15)\)

    15) \(\displaystyle (−6,8)\)

    Solution: \(\displaystyle (10,−0.927)(−10,−0.927+π)\)

    16) \(\displaystyle (4,3)\)

    17) \(\displaystyle (3,−\sqrt{3})\)

    Solution: \(\displaystyle 2\sqrt{3},−0.524)(−2\sqrt{3},−0.524+π)\)

    For the following exercises, find rectangular coordinates for the given point in polar coordinates.

    18) \(\displaystyle (2,\frac{5π}{4})\)

    19) \(\displaystyle (−2,\frac{π}{6})\)

    Solution: \(\displaystyle −\sqrt{3},−1)\)

    20) \(\displaystyle (5,\frac{π}{3})\)

    21) \(\displaystyle (1,\frac{7π}{6})\)

    Solution: \(\displaystyle (−\frac{\sqrt{3}}{2},\frac{−1}{2})\)

    22) \(\displaystyle (−3,\frac{3π}{4})\)

    23) \(\displaystyle (0,\frac{π}{2})\)

    Solution: \(\displaystyle (0,0)\)

    24) \(\displaystyle (−4.5,6.5)\)

    For the following exercises, determine whether the graphs of the polar equation are symmetric with respect to the \(\displaystyle x\)-axis, the \(\displaystyle y\) -axis, or the origin.

    25) \(\displaystyle r=3sin(2θ)\)

    Solution: Symmetry with respect to the x-axis, y-axis, and origin.

    26) \(\displaystyle r^2=9cosθ\)

    27) \(\displaystyle r=cos(\frac{θ}{5})\)

    Solution: Symmetric with respect to x-axis only.

    28) \(\displaystyle r=2secθ\)

    29) \(\displaystyle r=1+cosθ\)

    Solution: Symmetry with respect to x-axis only.

    For the following exercises, describe the graph of each polar equation. Confirm each description by converting into a rectangular equation.

    30) \(\displaystyle r=3\)

    31) \(\displaystyle θ=\frac{π}{4}\)

    Solution: Line \(\displaystyle y=x\)

    32) \(\displaystyle r=secθ\)

    33) \(\displaystyle r=cscθ\)

    Solution: \(\displaystyle y=1\)

    For the following exercises, convert the rectangular equation to polar form and sketch its graph.

    34) \(\displaystyle x^2+y^2=16\)

    35) \(\displaystyle x^2−y^2=16\)

    Solution: Hyperbola; polar form \(\displaystyle r^2cos(2θ)=16\) or \(\displaystyle r^2=16secθ.\)

    A hyperbola with vertices at (−4, 0) and (4, 0), the first pointing out into quadrants II and III and the second pointing out into quadrants I and IV.

    36) \(\displaystyle x=8\)

    For the following exercises, convert the rectangular equation to polar form and sketch its graph.

    37) \(\displaystyle 3x−y=2\)

    Solution: \(\displaystyle r=\frac{2}{3cosθ−sinθ}\)

    A straight line with slope 3 and y intercept −2.

    38) \(\displaystyle y^2=4x\)

    For the following exercises, convert the polar equation to rectangular form and sketch its graph.

    39) \(\displaystyle r=4sinθ\)

    40) \(\displaystyle x^2+y^2=4y\)

    Solution:

    A circle of radius 2 with center at (2, π/2).

    41) \(\displaystyle r=6cosθ\)

    42) \(\displaystyle r=θ\)

    Solution: \(\displaystyle xtan\sqrt{x^2+y^2}=y\)

    A spiral starting at the origin and crossing θ = π/2 between 1 and 2, θ = π between 3 and 4, θ = 3π/2 between 4 and 5, θ = 0 between 6 and 7, θ = π/2 between 7 and 8, and θ = π between 9 and 10.

    43) \(\displaystyle r=cotθcscθ\)

    For the following exercises, sketch a graph of the polar equation and identify any symmetry.

    44) \(\displaystyle r=1+sinθ\)

    Solution: y-axis symmetry

    A cardioid with the upper heart part at the origin and the rest of the cardioid oriented up.

    45) \(\displaystyle r=3−2cosθ\)

    46) \(\displaystyle r=2−2sinθ\)

    Solution: y-axis symmetry

    A cardioid with the upper heart part at the origin and the rest of the cardioid oriented down.

    47) \(\displaystyle r=5−4sinθ\)

    48) \(\displaystyle r=3cos(2θ)\)

    Solution: x- and y-axis symmetry and symmetry about the pole

    A rose with four petals that reach their furthest extent from the origin at θ = 0, π/2, π, and 3π/2.

    49) \(\displaystyle r=3sin(2θ)\)

    50) \(\displaystyle r=2cos(3θ)\)

    Solution: x-axis symmetry

    A rose with three petals that reach their furthest extent from the origin at θ = 0, 2π/3, and 4π/3.

    51) \(\displaystyle r=3cos(\frac{θ}{2})\)

    52) \(\displaystyle r^2=4cos(\frac{2}{θ})\)

    Solution: x- and y-axis symmetry and symmetry about the pole

    The infinity symbol with the crossing point at the origin and with the furthest extent of the two petals being at θ = 0 and π.

    53) \(\displaystyle r^2=4sinθ\)

    54) \(\displaystyle r=2θ\)

    Solution: no symmetry

    A spiral that starts at the origin crossing the line θ = π/2 between 3 and 4, θ = π between 6 and 7, θ = 3π/2 between 9 and 10, θ = 0 between 12 and 13, θ = π/2 between 15 and 16, and θ = π between 18 and 19.

    55) [T] The graph of \(\displaystyle r=2cos(2θ)sec(θ).\) is called a strophoid. Use a graphing utility to sketch the graph, and, from the graph, determine the asymptote.

    56) [T] Use a graphing utility and sketch the graph of \(\displaystyle r=\frac{6}{2sinθ−3cosθ}\).

    Solution: a line

    A line that crosses the y axis at roughly 3 and has slope roughly 3/2.

    57) [T] Use a graphing utility to graph \(\displaystyle r=\frac{1}{1−cosθ}\).

    58) [T] Use technology to graph \(\displaystyle r=e^{sin(θ)}−2cos(4θ)\).

    Solution:

    A geometric shape that resembles a butterfly with larger wings in the first and second quadrants, smaller wings in the third and fourth quadrants, a body along the θ = π/2 line and legs along the θ = 0 and π lines.

    59) [T] Use technology to plot \(\displaystyle r=sin(\frac{3θ}{7})\) (use the interval \(\displaystyle 0≤θ≤14π\)).

    60) Without using technology, sketch the polar curve \(\displaystyle θ=\frac{2π}{3}\).

    Solution:

    A line with θ = 120°.

    61) [T] Use a graphing utility to plot \(\displaystyle r=θsinθ\) for \(\displaystyle −π≤θ≤π\).

    62) [T] Use technology to plot \(\displaystyle r=e^{−0.1θ}\) for \(\displaystyle −10≤θ≤10.\)

    Solution:

    A spiral that starts in the third quadrant.

    63) [T] There is a curve known as the “Black Hole.” Use technology to plot \(\displaystyle r=e^{−0.01θ}\) for \(\displaystyle −100≤θ≤100\).

    64) [T] Use the results of the preceding two problems to explore the graphs of \(\displaystyle r=e^{−0.001θ}\) and \(\displaystyle r=e^{−0.0001θ}\) for \(\displaystyle |θ|>100\).

    Solution: Answers vary. One possibility is the spiral lines become closer together and the total number of spirals increases.

    11.4: Area and Arc Length in Polar Coordinates

    For the following exercises, determine a definite integral that represents the area.

    1) Region enclosed by \(\displaystyle r=4\)

    2) Region enclosed by \(\displaystyle r=3sinθ\)

    Solution: \(\displaystyle \frac{9}{2}∫^π_0sin^2θdθ\)

    3) Region in the first quadrant within the cardioid \(\displaystyle r=1+sinθ\)

    4) Region enclosed by one petal of \(\displaystyle r=8sin(2θ)\)

    Solution: \(\displaystyle \frac{3}{2}∫^{π/2}_0sin^2(2θ)dθ\)

    5) Region enclosed by one petal of \(\displaystyle r=cos(3θ)\)

    6) Region below the polar axis and enclosed by \(\displaystyle r=1−sinθ\)

    Solution: \(\displaystyle \frac{1}{2}∫^{2π}_π(1−sinθ)^2dθ\)

    7) Region in the first quadrant enclosed by \(\displaystyle r=2−cosθ\)

    8) Region enclosed by the inner loop of \(\displaystyle r=2−3sinθ\)

    Solution: \(\displaystyle ∫^{π/2}_{sin^{−1}(2/3)}(2−3sinθ)^2dθ\)

    9) Region enclosed by the inner loop of \(\displaystyle r=3−4cosθ\)

    10) Region enclosed by \(\displaystyle r=1−2cosθ\) and outside the inner loop

    Solution: \(\displaystyle ∫^π_0(1−2cosθ)^2dθ−∫^{π/3}_0(1−2cosθ)^2dθ\)

    11) Region common to \(\displaystyle r=3sinθ\) and \(\displaystyle r=2−sinθ\)

    12) Region common to \(\displaystyle r=2\) and \(\displaystyle r=4cosθ\)

    Solution: \(\displaystyle 4∫^{π/3}_0dθ+16∫^{π/2}_{π/3}(cos^2θ)dθ\)

    13) Region common to \(\displaystyle r=3cosθ\) and \(\displaystyle r=3sinθ\)

    For the following exercises, find the area of the described region.

    14) Enclosed by \(\displaystyle r=6sinθ\)

    Solution: \(\displaystyle 9π\)

    15) Above the polar axis enclosed by \(\displaystyle r=2+sinθ\)

    16) Below the polar axis and enclosed by \(\displaystyle r=2−cosθ\)

    Solution: \(\displaystyle \frac{9π}{4}\)

    17) Enclosed by one petal of \(\displaystyle r=4cos(3θ)\)

    18) Enclosed by one petal of \(\displaystyle r=3cos(2θ)\)

    Solution: \(\displaystyle \frac{9π}{8}\)

    19) Enclosed by \(\displaystyle r=1+sinθ\)

    20) Enclosed by the inner loop of \(\displaystyle r=3+6cosθ\)

    Solution: \(\displaystyle \frac{18π−27\sqrt{3}}{2}\)

    21) Enclosed by \(\displaystyle r=2+4cosθ\) and outside the inner loop

    22) Common interior of \(\displaystyle r=4sin(2θ)\) and \(\displaystyle r=2\)

    Solution: \(\displaystyle \frac{4}{3}(4π−3\sqrt{3})\)

    23) Common interior of \(\displaystyle r=3−2sinθ\) and \(\displaystyle r=−3+2sinθ\)

    24) Common interior of \(\displaystyle r=6sinθ\) and \(\displaystyle r=3\)

    Solution: \(\displaystyle 32(4π−33√)

    25) Inside \(\displaystyle r=1+cosθ\) and outside \(\displaystyle r=cosθ\)

    26) Common interior of \(\displaystyle r=2+2cosθ\) and \(\displaystyle r=2sinθ\)

    Solution: \(\displaystyle 2π−4\)

    For the following exercises, find a definite integral that represents the arc length.

    27) \(\displaystyle r=4cosθ\) on the interval \(\displaystyle 0≤θ≤\frac{π}{2}\)

    28) \(\displaystyle r=1+sinθ\) on the interval \(\displaystyle 0≤θ≤2π\)

    Solution: \(\displaystyle ∫^{2π}_0\sqrt{(1+sinθ)^2+cos^2θ}dθ\)

    29) \(\displaystyle r=2secθ\) on the interval \(\displaystyle 0≤θ≤\frac{π}{3}\)

    30) \(\displaystyle r=e^θ\) on the interval \(\displaystyle 0≤θ≤1\)

    Solution: \(\displaystyle \sqrt{2}∫^1_0e^θdθ\)

    For the following exercises, find the length of the curve over the given interval.

    31) \(\displaystyle r=6\) on the interval \(\displaystyle 0≤θ≤\frac{π}{2}\)

    32) \(\displaystyle r=e^{3θ}\) on the interval \(\displaystyle 0≤θ≤2\)

    Solution: \(\displaystyle \frac{\sqrt{10}}{3}(e^6−1)\)

    33) \(\displaystyle r=6cosθ\) on the interval \(\displaystyle 0≤θ≤\frac{π}{2}\)

    34) \(\displaystyle r=8+8cosθ\) on the interval \(\displaystyle 0≤θ≤π\)

    Solution: 32

    35) \(\displaystyle r=1−sinθ\) on the interval \(\displaystyle 0≤θ≤2π\)

    For the following exercises, use the integration capabilities of a calculator to approximate the length of the curve.

    36) [T] \(\displaystyle r=3θ\) on the interval \(\displaystyle 0≤θ≤\frac{π}{2}\)

    Solution: 6.238

    37) [T] \(\displaystyle r=\frac{2}{θ}\) on the interval \(\displaystyle π≤θ≤2π\)

    38) [T] \(\displaystyle r=sin^2(\frac{θ}{2})\) on the interval \(\displaystyle 0≤θ≤π\)

    Solution: 2

    39) [T] \(\displaystyle r=2θ^2\) on the interval \(\displaystyle 0≤θ≤π\)

    40) [T] \(\displaystyle r=sin(3cosθ)\) on the interval \(\displaystyle 0≤θ≤π\)

    Solution: 4.39

    For the following exercises, use the familiar formula from geometry to find the area of the region described and then confirm by using the definite integral.

    41) \(\displaystyle r=3sinθ\) on the interval \(\displaystyle 0≤θ≤π\)

    42) \(\displaystyle r=sinθ+cosθ\) on the interval \(\displaystyle 0≤θ≤π\)

    Solution: \(\displaystyle A=π(\frac{\sqrt{2}}{2})^2=\frac{π}{2}\) and \(\displaystyle \frac{1}{2}∫^π_0(1+2sinθcosθ)dθ=\frac{π}{2}\)

    43) \(\displaystyle r=6sinθ+8cosθ\) on the interval \(\displaystyle 0≤θ≤π\)

    For the following exercises, use the familiar formula from geometry to find the length of the curve and then confirm using the definite integral.

    44) \(\displaystyle r=3sinθ\) on the interval \(\displaystyle 0≤θ≤π\)

    Solution: \(\displaystyle C=2π(\frac{3}{2})=3π\) and \(\displaystyle ∫^π_03dθ=3π\)

    45) \(\displaystyle r=sinθ+cosθ\) on the interval \(\displaystyle 0≤θ≤π\)

    46) \(\displaystyle r=6sinθ+8cosθ\) on the interval \(\displaystyle 0≤θ≤π\)

    Solution: \(\displaystyle C=2π(5)=10π\) and \(\displaystyle ∫^π_010dθ=10π\)

    47) Verify that if \(\displaystyle y=rsinθ=f(θ)sinθ\) then \(\displaystyle \frac{dy}{dθ}=f'(θ)sinθ+f(θ)cosθ.\)

    For the following exercises, find the slope of a tangent line to a polar curve \(\displaystyle r=f(θ)\). Let \(\displaystyle x=rcosθ=f(θ)cosθ\) and \(\displaystyle y=rsinθ=f(θ)sinθ\), so the polar equation \(\displaystyle r=f(θ)\) is now written in parametric form.

    48) Use the definition of the derivative \(\displaystyle dydx=\frac{dy/dθ}{dx/dθ}\) and the product rule to derive the derivative of a polar equation.

    Solution: \(\displaystyle \frac{dy}{dx}=\frac{f′(θ)sinθ+f(θ)cosθ}{f′(θ)cosθ−f(θ)sinθ}\)

    49) \(\displaystyle r=1−sinθ; (\frac{1}{2},\frac{π}{6})\)

    50) \(\displaystyle r=4cosθ; (2,\frac{π}{3})\)

    Solution: The slope is \(\displaystyle \frac{1}{\sqrt{3}}\).

    51) \(\displaystyle r=8sinθ; (4,\frac{5π}{6})\)

    52) \(\displaystyle r=4+sinθ; (3,\frac{3π}{2})\)

    Solution: The slope is 0.

    53) \(\displaystyle r=6+3cosθ; (3,π)\)

    54) \(\displaystyle r=4cos(2θ);\) tips of the leaves

    Solution: At \(\displaystyle (4,0),\) the slope is undefined. At \(\displaystyle (−4,\frac{π}{2})\), the slope is 0.

    55) \(\displaystyle r=2sin(3θ);\) tips of the leaves

    56) \(\displaystyle r=2θ; (\frac{π}{2},\frac{π}{4})\)

    Solution: The slope is undefined at \(\displaystyle θ=\frac{π}{4}\).

    57) Find the points on the interval \(\displaystyle −π≤θ≤π\) at which the cardioid \(\displaystyle r=1−cosθ\) has a vertical or horizontal tangent line.

    58) For the cardioid \(\displaystyle r=1+sinθ,\) find the slope of the tangent line when \(\displaystyle θ=\frac{π}{3}\).

    Solution: Slope = −1.

    For the following exercises, find the slope of the tangent line to the given polar curve at the point given by the value of \(\displaystyle θ\).

    59) \(\displaystyle r=3cosθ,θ=\frac{π}{3}\)

    60) \(\displaystyle r=θ, θ=\frac{π}{2}\)

    Soltuion: Slope is \(\displaystyle \frac{−2}{π}\).

    61) \(\displaystyle r=lnθ, θ=e\)

    62) [T] Use technology: \(\displaystyle r=2+4cosθ\) at \(\displaystyle θ=\frac{π}{6}\)

    Solution: Calculator answer: −0.836.

    For the following exercises, find the points at which the following polar curves have a horizontal or vertical tangent line.

    63) \(\displaystyle r=4cosθ\)

    64) \(\displaystyle r^2=4cos(2θ)\)

    Solution: Horizontal tangent at \(\displaystyle (±\sqrt{2},\frac{π}{6}), (±\sqrt{2},−\frac{π}{6})\).

    65) \(\displaystyle r=2sin(2θ)\)

    66) The cardioid \(\displaystyle r=1+sinθ\)

    Solution: Horizontal tangents at \(\displaystyle \frac{π}{2},\frac{7π}{6},\frac{11π}{6}.\) Vertical tangents at \(\displaystyle \frac{π}{6},\frac{5π}{6}\) and also at the pole \(\displaystyle (0,0)\).

    67) Show that the curve \(\displaystyle r=sinθtanθ\) (called a cissoid of Diocles) has the line \(\displaystyle x=1\) as a vertical asymptote.

    11.5: Conic Sections

    For the following exercises, determine the equation of the parabola using the information given.

    1) Focus \(\displaystyle (4,0)\) and directrix \(\displaystyle x=−4\)

    Solution: \(\displaystyle y^2=16x\)

    2) Focus \(\displaystyle (0,−3)\) and directrix \(\displaystyle y=3\)

    3) Focus \(\displaystyle (0,0.5)\) and directrix \(\displaystyle y=−0.5\)

    Solution: \(\displaystyle x^2=2y\)

    4) Focus \(\displaystyle (2,3)\) and directrix \(\displaystyle x=−2\)

    5) Focus \(\displaystyle (0,2)\) and directrix \(\displaystyle y=4\)

    Solution: \(\displaystyle x^2=−4(y−3)\)

    6) Focus \(\displaystyle (−1,4)\) and directrix \(\displaystyle x=5\)

    7) Focus \(\displaystyle (−3,5)\) and directrix \(\displaystyle y=1\)

    Solution: \(\displaystyle (x+3)^2=8(y−3)\)

    8) Focus \(\displaystyle (\frac{5}{2},−4)\) and directrix \(\displaystyle x=\frac{7}{2}\)

    For the following exercises, determine the equation of the ellipse using the information given.

    9) Endpoints of major axis at \(\displaystyle (4,0),(−4,0)\) and foci located at \(\displaystyle (2,0),(−2,0)\)

    Solution: \(\displaystyle \frac{x^2}{16}+\frac{y^2}{12}=1\)

    10) Endpoints of major axis at \(\displaystyle (0,5),(0,−5)\) and foci located at \(\displaystyle (0,3),(0,−3)\)

    11) Endpoints of major axis at \(\displaystyle (0,2),(0,−2)\) and foci located at \(\displaystyle (3,0),(−3,0)\)

    Solution: \(\displaystyle \frac{x^2}{13}+\frac{y^2}{4}=1\)

    12) Endpoints of major axis at \(\displaystyle (−3,3),(7,3)\) and foci located at \(\displaystyle (−2,3),(6,3)\)

    13) Endpoints of major axis at \(\displaystyle (−3,5),(−3,−3)\) and foci located at \(\displaystyle (−3,3),(−3,−1)\)

    Solution: \(\displaystyle \frac{(y−1)^2}{16}+\frac{(x+3)^2}{12}=1\)

    14) Endpoints of major axis at \(\displaystyle (0,0),(0,4)\) and foci located at \(\displaystyle (5,2),(−5,2)\)

    15) Foci located at \(\displaystyle (2,0),(−2,0)\) and eccentricity of \(\displaystyle \frac{1}{2}\)

    Solutin: \(\displaystyle \frac{x^2}{16}+\frac{y^2}{12}=1\)

    16) Foci located at \(\displaystyle (0,−3),(0,3)\) and eccentricity of \(\displaystyle \frac{3}{4}\)

    For the following exercises, determine the equation of the hyperbola using the information given.

    17) Vertices located at \(\displaystyle (5,0),(−5,0)\) and foci located at \(\displaystyle (6,0),(−6,0)\)

    Solution: \(\displaystyle \frac{x^2}{25}−\frac{y^2}{11}=1\)

    18) Vertices located at \(\displaystyle (0,2),(0,−2)\) and foci located at \(\displaystyle (0,3),(0,−3)\)

    19) Endpoints of the conjugate axis located at \(\displaystyle (0,3),(0,−3)\) and foci located \(\displaystyle (4,0),(−4,0)\)

    Solution: \(\displaystyle \frac{x^2}{7}−\frac{y^2}{9}=1\)

    20) Vertices located at \(\displaystyle (0,1),(6,1)\) and focus located at \(\displaystyle (8,1)\)

    21) Vertices located at \(\displaystyle (−2,0),(−2,−4)\) and focus located at \(\displaystyle (−2,−8)\)

    Solution: \(\displaystyle \frac{(y+2)^2}{4}−\frac{(x+2)^2}{32}=1\)

    22) Endpoints of the conjugate axis located at \(\displaystyle (3,2),(3,4)\) and focus located at \(\displaystyle (3,7)\)

    23) Foci located at \(\displaystyle (6,−0),(6,0)\) and eccentricity of \(\displaystyle 3\)

    Soluton: \(\displaystyle \frac{x^2}{4}−\frac{y^2}{32}=1\)

    24) \(\displaystyle (0,10),(0,−10)\) and eccentricity of 2.5

    For the following exercises, consider the following polar equations of conics. Determine the eccentricity and identify the conic.

    25) \(\displaystyle r=\frac{−1}{1+cosθ}\)

    Solution: \(\displaystyle e=1,\) parabola

    26) \(\displaystyle r=\frac{8}{2−sinθ}\)

    27) \(\displaystyle r=\frac{5}{2+sinθ}\)

    Solution: \(\displaystyle e=\frac{1}{2},\) ellipse

    28) \(\displaystyle r=\frac{5}{−1+2sinθ}\)

    29) \(\displaystyle r=\frac{3}{2−6sinθ}\)

    Solution: \(\displaystyle e=3\), hyperbola

    30) \(\displaystyle r=\frac{3}{−4+3sinθ}\)

    For the following exercises, find a polar equation of the conic with focus at the origin and eccentricity and directrix as given.

    31) Directrix: \(\displaystyle x=4;e=\frac{1}{5}\)

    Solution: \(\displaystyle r=\frac{4}{5+cosθ}\)

    32) Directrix: \(\displaystyle x=−4;e=5\)

    33) Directrix: \(\displaystyle y=2;e=2\)

    Solution: \(\displaystyle r=\frac{4}{1+2sinθ}\)

    34) Directrix: \(\displaystyle y=−2;e=\frac{1}{2}\)

    For the following exercises, sketch the graph of each conic.

    35) \(\displaystyle r=\frac{1}{1+sinθ}\)

    Solution:

    Graph of a parabola open down with center at the origin.

    36) \(\displaystyle r=\frac{1}{1−cosθ}\)

    37) \(\displaystyle r=\frac{4}{1+cosθ}\)

    Solution:

    Graph of a parabola open to the left with center near the origin.

    38) \(\displaystyle r=\frac{10}{5+4sinθ}\)

    39) \(\displaystyle r=\frac{15}{3−2cosθ}\)

    Soluton:

    Graph of an ellipse with center near (8, 0), major axis horizontal and roughly 18, and minor axis slightly more than 12.

    40) \(\displaystyle r=\frac{32}{3+5sinθ}\)

    41) \(\displaystyle r(2+sinθ)=4\)

    Solution:

    Graph of an circle with center near (0, −1.5) and radius near 2.5.

    42) \(\displaystyle r=\frac{3}{2+6sinθ}\)

    43) \(\displaystyle r=\frac{3}{−4+2sinθ}\)

    Solution:

    Graph of a circle with center (0, −0.5) and radius 1.

    44) \(\displaystyle \frac{x^2}{9}+\frac{y^2}{4}=1\)

    45) \(\displaystyle \frac{x^2}{4}+\frac{y^2}{16}=1\)

    Solution:

    Graph of an ellipse with center the origin and with major axis vertical and of length 8 and minor axis of length 4.

    46) \(\displaystyle 4x^2+9y^2=36\)

    47) \(\displaystyle 25x^2−4y^2=100\)

    Solution:

    Graph of a hyperbola with center the origin and with the two halves open to the left and right. The vertices are on the x axis at ±2.

    48) \(\displaystyle \frac{x^2}{16}−\frac{y^2}{9}=1\)

    49) \(\displaystyle x^2=12y\)

    Solution:

    Graph of a parabola with vertex the origin and open up.

    50) \(\displaystyle y^2=20x\)

    51) \(\displaystyle 12x=5y^2\)

    Solution:

    Graph of a parabola with vertex the origin and open to the right.

    For the following equations, determine which of the conic sections is described.

    52) \(\displaystyle xy=4\)

    53) \(\displaystyle x^2+4xy−2y^2−6=0\)

    Solution: Hyperbola

    54) \(\displaystyle x^2+2\sqrt{3}xy+3y^2−6=0\)

    55) \(\displaystyle x^2−xy+y^2−2=0\)

    Solution: Ellipse

    56) \(\displaystyle 34x^2−24xy+41y^2−25=0\)

    57) \(\displaystyle 52x^2−72xy+73y^2+40x+30y−75=0\)

    Solution: Ellipse

    58) The mirror in an automobile headlight has a parabolic cross section, with the lightbulb at the focus. On a schematic, the equation of the parabola is given as \(\displaystyle x^2=4y\). At what coordinates should you place the lightbulb?

    59) A satellite dish is shaped like a paraboloid of revolution. The receiver is to be located at the focus. If the dish is 12 feet across at its opening and 4 feet deep at its center, where should the receiver be placed?

    Solution: At the point 2.25 feet above the vertex.

    60) Consider the satellite dish of the preceding problem. If the dish is 8 feet across at the opening and 2 feet deep, where should we place the receiver?

    61) A searchlight is shaped like a paraboloid of revolution. A light source is located 1 foot from the base along the axis of symmetry. If the opening of the searchlight is 3 feet across, find the depth.

    Solution: 0.5625 feet

    62) Whispering galleries are rooms designed with elliptical ceilings. A person standing at one focus can whisper and be heard by a person standing at the other focus because all the sound waves that reach the ceiling are reflected to the other person. If a whispering gallery has a length of 120 feet and the foci are located 30 feet from the center, find the height of the ceiling at the center.

    63) A person is standing 8 feet from the nearest wall in a whispering gallery. If that person is at one focus and the other focus is 80 feet away, what is the length and the height at the center of the gallery?

    Solution: Length is 96 feet and height is approximately 26.53 feet.

    For the following exercises, determine the polar equation form of the orbit given the length of the major axis and eccentricity for the orbits of the comets or planets. Distance is given in astronomical units (AU).

    64) Halley’s Comet: length of major axis = 35.88, eccentricity = 0.967

    65) Hale-Bopp Comet: length of major axis = 525.91, eccentricity = 0.995

    Solution: \(\displaystyle r=\frac{2.616}{1+0.995cosθ}\)

    66) Mars: length of major axis = 3.049, eccentricity = 0.0934

    67) Jupiter: length of major axis = 10.408, eccentricity = 0.0484

    Solution: \(\displaystyle r=\frac{5.192}{1+0.0484cosθ}\)

    Chapter Review Exercises

    True or False? Justify your answer with a proof or a counterexample.

    1) The rectangular coordinates of the point \(\displaystyle (4,\frac{5π}{6})\) are \(\displaystyle (2\sqrt{3},−2).\)

    2) The equations \(\displaystyle x=cosh(3t), y=2sinh(3t)\) represent a hyperbola.

    Solution: True.

    3) The arc length of the spiral given by \(\displaystyle r=\frac{θ}{2}\) for \(\displaystyle 0≤θ≤3π\) is \(\displaystyle \frac{9}{4}π^3\).

    4) Given \(\displaystyle x=f(t)\) and \(\displaystyle y=g(t)\), if \(\displaystyle \frac{dx}{dy}=\frac{dy}{dx}\), then \(\displaystyle f(t)=g(t)+C,\) where \(\displaystyle C\) is a constant.

    Solution: False. Imagine \(\displaystyle y=t+1, x=−t+1.\)

    For the following exercises, sketch the parametric curve and eliminate the parameter to find the Cartesian equation of the curve.

    5) \(\displaystyle x=1+t, y=t^2−1, −1≤t≤1\)

    6) \(\displaystyle x=e^t, y=1−e^{3t}, 0≤t≤1\)

    Solution: \(\displaystyle y=1−x^3\)

    Graph of a curve starting at (1, 0) and decreasing into the fourth quadrant.

    7) \(\displaystyle x=sinθ, y=1−cscθ, 0≤θ≤2π\)

    8) \(\displaystyle x=4cosϕ, y=1−sinϕ, 0≤ϕ≤2π\)

    Solution: \(\displaystyle \frac{x^2}{16}+(y−1)^2=1\)

    Graph of an ellipse with center (0, 1), major axis horizontal and of length 8, and minor axis of length 2.

    For the following exercises, sketch the polar curve and determine what type of symmetry exists, if any.

    9) \(\displaystyle r=4sin(\frac{θ}{3})\)

    10) \(\displaystyle r=5cos(5θ)\)

    Solution: Symmetric about polar axis

    Graph of a five-petaled rose with initial petal at θ = 0.

    For the following exercises, find the polar equation for the curve given as a Cartesian equation.

    11) \(\displaystyle x+y=5\)

    12) \(\displaystyle y^2=4+x^2\)

    Solution: \(\displaystyle r^2=\frac{4}{sin^2θ−cos^2θ}\)

    For the following exercises, find the equation of the tangent line to the given curve. Graph both the function and its tangent line.

    13) \(\displaystyle x=ln(t), y=t^2−1, t=1\)

    14) \(\displaystyle r=3+cos(2θ), θ=\frac{3π}{4}\)

    Solution: \(\displaystyle y=\frac{3\sqrt{2}}{2}+\frac{1}{5}(x+\frac{3\sqrt{2}}{2})\)

    Graph of a peanut-shaped figure, with y intercepts at ±2 and x intercepts at ±4. The tangent line occurs in the second quadrant.

    15) Find \(\displaystyle \frac{dy}{dx}, \frac{dx}{dy},\) and \(\displaystyle \frac{d^2x}{dy^2}\) of \(\displaystyle y=(2+e^{−t}), x=1−sin(t)\)

    For the following exercises, find the area of the region.

    16) \(\displaystyle x=t^2, y=ln(t), 0≤t≤e\)

    Solution: \(\displaystyle \frac{e^2}{2}\)

    17) \(\displaystyle r=1−sinθ\) in the first quadrant

    For the following exercises, find the arc length of the curve over the given interval.

    18) \(\displaystyle x=3t+4, y=9t−2, 0≤t≤3\)

    Solution: \(\displaystyle 9\sqrt{10}\)

    19) \(\displaystyle r=6cosθ, 0≤θ≤2π.\) Check your answer by geometry.

    For the following exercises, find the Cartesian equation describing the given shapes.

    20) A parabola with focus \(\displaystyle (2,−5)\) and directrix \(\displaystyle x=6\)

    Solution: \(\displaystyle (y+5)^2=−8x+32\)

    21) An ellipse with a major axis length of 10 and foci at \(\displaystyle (−7,2)\) and \(\displaystyle (1,2)\)

    22) A hyperbola with vertices at \(\displaystyle (3,−2)\) and \(\displaystyle (−5,−2)\) and foci at \(\displaystyle (−2,−6)\) and \(\displaystyle (−2,4)\)

    Solution: \(\displaystyle \frac{(y+1)^2}{16}−\frac{(x+2)^2}{9}=1\)

    For the following exercises, determine the eccentricity and identify the conic. Sketch the conic.

    23) \(\displaystyle r=\frac{6}{1+3cos(θ)}\)

    24) \(\displaystyle r=\frac{4}{3−2cosθ}\)

    Solution: \(\displaystyle e=\frac{2}{3}\), ellipse

    Graph of an ellipse with center near (1.5, 0), major axis nearly 5 and horizontal, and minor axis nearly 4.

    25) \(\displaystyle r=\frac{7}{5−5cosθ}\)

    26) Determine the Cartesian equation describing the orbit of Pluto, the most eccentric orbit around the Sun. The length of the major axis is 39.26 AU and minor axis is 38.07 AU. What is the eccentricity?

    Solution: \(\displaystyle \frac{y^2}{19.03^2}+\frac{x^2}{19.63^2}=1, e=0.2447\)

    27) The C/1980 E1 comet was observed in 1980. Given an eccentricity of 1.057 and a perihelion (point of closest approach to the Sun) of 3.364 AU, find the Cartesian equations describing the comet’s trajectory. Are we guaranteed to see this comet again? (Hint: Consider the Sun at point \(\displaystyle (0,0)\).)