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Mathematics LibreTexts

4.5E:

  • Page ID
    18600
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    The odd-numbered problems have answers.

    Exercise \(\PageIndex{1}\) Equation of Parabola

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

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

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

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

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

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

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

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

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

    Answer

    Solution 1: \(\displaystyle y^2=16x\), Solution 3: \(\displaystyle x^2=2y\), Solution 5: \(\displaystyle x^2=−4(y−3)\), Solution 7: \(\displaystyle (x+3)^2=8(y−3)\)

     

    Exercise \(\PageIndex{2}\) Ellipse

    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 \(\displaystyle (2,0),(−2,0)\)

    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 \(\displaystyle (3,0),(−3,0)\)

    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 \(\displaystyle (−3,3),(−3,−1)\)

    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 \(\displaystyle \frac{1}{2}\)

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

    Answer

    Solution 9: \(\displaystyle \frac{x^2}{16}+\frac{y^2}{12}=1\), Solution 11: \(\displaystyle \frac{x^2}{13}+\frac{y^2}{4}=1\), Solution 13: \(\displaystyle \frac{(y−1)^2}{16}+\frac{(x+3)^2}{12}=1\), Solution 15: \(\displaystyle \frac{x^2}{16}+\frac{y^2}{12}=1\)

    Exercise \(\PageIndex{3}\) Hyperbola

    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 \(\displaystyle (6,0),(−6,0)\)

    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 \(\displaystyle (4,0),(−4,0)\)

    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 \(\displaystyle (−2,−8)\)

    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 \(\displaystyle 3\)

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

    Answer

    Solution 17: \(\displaystyle \frac{x^2}{25}−\frac{y^2}{11}=1\), Solution 19: \(\displaystyle \frac{x^2}{7}−\frac{y^2}{9}=1\), Solution 21: \(\displaystyle \frac{(y+2)^2}{4}−\frac{(x+2)^2}{32}=1\), Soluton 23: \(\displaystyle \frac{x^2}{4}−\frac{y^2}{32}=1\)

    Exercise \(\PageIndex{4}\)  identify the conic

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

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

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

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

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

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

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

    Answer

    Solution 25: \(\displaystyle e=1,\) parabola, Solution 27: \(\displaystyle e=\frac{1}{2},\) ellipse, Solution 29: \(\displaystyle e=3\), hyperbola, 

    Exercise \(\PageIndex{5}\) Polar equation of the conic

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

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

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

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

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

    Answer

    Solution 31: \(\displaystyle r=\frac{4}{5+cosθ}\), Solution 33: \(\displaystyle r=\frac{4}{1+2sinθ}\)

     

    Exercise \(\PageIndex{6}\) Sketch the graph

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

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

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

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

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

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

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

    41) \(\displaystyle r(2+sinθ)=4\)

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

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

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

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

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

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

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

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

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

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

     

    Answer

    Solution 35:

    Solution 37:

    solution 39:

     

    Solution 41:

    Solution 43:

    Solution 45:

     

    Solution 47:

    Solution 49:

    Solution 51:

     

     

    Exercise \(\PageIndex{7}\) Conic Sections

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

    52) \(\displaystyle xy=4\)

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

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

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

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

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

     

    Answer

    Solution 53: Hyperbola, Solution 55: Ellipse, Solution 57: Ellipse

    Exercise \(\PageIndex{8}\) Applications

    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?

    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.

    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?

     

    Answer

    Solution 59: At the point 2.25 feet above the vertex, Solution 61: 0.5625 feet, Solution 63: Length is 96 feet and height is approximately 26.53 feet.

    Exercise \(\PageIndex{9}\) Polar Form

    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

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

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

    Answer

    Solution 65: \(\displaystyle r=\frac{2.616}{1+0.995cosθ}\), Solution 67: \(\displaystyle r=\frac{5.192}{1+0.0484cosθ}\).