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8.E: Conic Sections (Exercises)

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    Exercise \(\PageIndex{1}\)

    Calculate the distance and midpoint between the given two points.

    1. \((0,2)\) and \((-4,-1)\)
    2. \((6,0)\) and \((-2,-6)\)
    3. \((-2,4)\) and \((-6,-8)\)
    4. \(\left(\frac{1}{2},-1\right)\) and \(\left(\frac{5}{2},-\frac{1}{2}\right)\)
    5. \((0,-3 \sqrt{2})\) and \((\sqrt{5},-4 \sqrt{2})\)
    6. \((-5 \sqrt{3}, \sqrt{6})\) and \((-3 \sqrt{3}, \sqrt{6})\)
    Answer

    1. Distance: \(5\) units; midpoint: \(\left(-2, \frac{1}{2}\right)\)

    3. Distance: \(4\sqrt{10}\) units; midpoint: \((-4,-2)\)

    5. Distance: \(\sqrt{7}\) units; midpoint: \(\left(\frac{\sqrt{5}}{2},-\frac{7 \sqrt{2}}{2}\right)\)

    Exercise \(\PageIndex{2}\)

    Determine the area of a circle whose diameter is defined by the given two points.

    1. \((-3,3)\) and \((3,-3)\)
    2. \((-2,-9)\) and \((-10,-15)\)
    3. \(\left(\frac{2}{3},-\frac{1}{2}\right)\) and \(\left(-\frac{1}{3}, \frac{3}{2}\right)\)
    4. \((2 \sqrt{5},-2 \sqrt{2})\) and \((0,-4 \sqrt{2})\)
    Answer

    1. \(18\pi\) square units

    3. \(\frac{5 \pi}{4}\) square units

    Exercise \(\PageIndex{3}\)

    Rewrite in standard form and give the vertex.

    1. \(y=x^{2}-10 x+33\)
    2. \(y=2 x^{2}-4 x-1\)
    3. \(y=x^{2}-3 x-1\)
    4. \(y=-x^{2}-x-2\)
    5. \(x=y^{2}+10 y+10\)
    6. \(x=3 y^{2}+12 y+7\)
    7. \(x=-y^{2}+8 y-3\)
    8. \(x=5 y^{2}-5 y+2\)
    Answer

    1. \(y=(x-5)^{2}+8 ;\) vertex: \((5,8)\)

    3. \(y=\left(x-\frac{3}{2}\right)^{2}-\frac{13}{4} ;\) vertex: \(\left(\frac{3}{2},-\frac{13}{4}\right)\)

    5. \(x=(y+5)^{2}-15 ;\) vertex: \((-15,-5)\)

    7. \(x=-(y-4)^{2}+13 ;\) vertex: \((13,4)\)

    Exercise \(\PageIndex{4}\)

    Rewrite in standard form and graph. Be sure to find the vertex and all intercepts.

    1. \(y=x^{2}-20 x+75\)
    2. \(y=-x^{2}-10 x+75\)
    3. \(y=-2 x^{2}-12 x-24\)
    4. \(y=4 x^{2}+4 x+6\)
    5. \(x=y^{2}-10 y+16\)
    6. \(x=-y^{2}+4 y+12\)
    7. \(x=-4 y^{2}+12 y\)
    8. \(x=9 y^{2}+18 y+12\)
    9. \(x=-4 y^{2}+4 y+2\)
    10. \(x=-y^{2}-5 y+2\)
    Answer

    1. \(y=(x-10)^{2}-25\);

    897c1db754e7a1fd038af05b6edee1a5.png
    Figure 8.E.1

    3. \(y=-2(x+3)^{2}-6\);

    4c5f575d96ad71dfaeaf14e720e4a33b.png
    Figure 8.E.2

    5. \(x=(y-5)^{2}-9\);

    e83f649f71ebd05f11d4c97a5f7992b1.png
    Figure 8.E.3

    7. \(x=-4\left(y-\frac{3}{2}\right)^{2}+9\);

    663ae0a27b5607cc3c234ff842fce2ad.png
    Figure 8.E.4

    9. \(x=-4\left(y-\frac{1}{2}\right)^{2}+3\);

    2e8569519d7b31c81ee38d7855d91e9c.png
    Figure 8.E.5

    Exercise \(\PageIndex{5}\)

    Determine the center and radius given the equation of a circle in standard form.

    1. \((x-6)^{2}+y^{2}=9\)
    2. \((x+8)^{2}+(y-10)^{2}=1\)
    3. \(x^{2}+y^{2}=5\)
    4. \(\left(x-\frac{3}{8}\right)^{2}+\left(y+\frac{5}{2}\right)^{2}=\frac{1}{2}\)
    Answer

    1. Center: \((6,0) ;\) radius: \(r=3\)

    3. Center: \((0,0) ;\) radius: \(r=\sqrt{5}\)

    Exercise \(\PageIndex{6}\)

    Determine standard form for the equation of the circle:

    1. Center \((-7,2)\) with radius \(r=10\)
    2. Center \(\left(\frac{1}{3},-1\right)\) with radius \(r=\frac{2}{3}\)
    3. Center \((0,-5)\) with radius \(r=2 \sqrt{7}\)
    4. Center \((1,0)\) with radius \(r=\frac{5 \sqrt{3}}{2}\)
    5. Circle whose diameter is defined by \((-4,10)\) and \((-2,8)\)
    6. Circle whose diameter is defined by \((3,-6)\) and \((0,-4)\)
    Answer

    1. \((x+7)^{2}+(y-2)^{2}=100\)

    3. \(x^{2}+(y+5)^{2}=28\)

    5. \((x+3)^{2}+(y-9)^{2}=2\)

    Exercise \(\PageIndex{7}\)

    Find the \(x\)- and \(y\)-intercepts.

    1. \((x-3)^{2}+(y+5)^{2}=16\)
    2. \((x+5)^{2}+(y-1)^{2}=4\)
    3. \(x^{2}+(y-2)^{2}=20\)
    4. \((x-3)^{2}+(y+3)^{2}=8\)
    5. \(x^{2}+y^{2}-12 y+27=0\)
    6. \(x^{2}+y^{2}-4 x+2 y+1=0\)
    Answer

    1. \(x\)-intercepts: none; \(y\)-intercepts: \((0,-5 \pm \sqrt{7})\)

    3. \(x\)-intercepts: \((\pm 4,0)\); \(y\)-intercepts: \((0,2 \pm 2 \sqrt{5})\)

    5. \(x\)-intercepts: none; \(y\)-intercepts: \((0,3),(0,9)\)

    Exercise \(\PageIndex{8}\)

    Graph.

    1. \((x+8)^{2}+(y-6)^{2}=4\)
    2. \((x-20)^{2}+\left(y+\frac{15}{2}\right)^{2}=\frac{225}{4}\)
    3. \(x^{2}+y^{2}=24\)
    4. \((x-1)^{2}+y^{2}=\frac{1}{4}\)
    5. \(x^{2}+(y-7)^{2}=27\)
    6. \((x+1)^{2}+(y-1)^{2}=2\)
    Answer

    1.

    4887782fe9f78e134a5596edb240d9f2.png
    Figure 8.E.6

    3.

    0c0ff6d02009afdff333ca8dc3d74740.png
    Figure 8.E.7

    5.

    f4772183b680f3cb2a33299c161ca2c1.png
    Figure 8.E.8

    Exercise \(\PageIndex{9}\)

    Rewrite in standard form and graph.

    1. \(x^{2}+y^{2}-6 x+4 y-3=0\)
    2. \(x^{2}+y^{2}+8 x-10 y+16=0\)
    3. \(2 x^{2}+2 y^{2}-2 x-6 y-3=0\)
    4. \(4 x^{2}+4 y^{2}+8 y+1=0\)
    5. \(x^{2}+y^{2}-5 x+y-\frac{1}{2}=0\)
    6. \(x^{2}+y^{2}+12 x-8 y=0\)
    Answer

    1. \((x-3)^{2}+(y+2)^{2}=16\);

    eeed6c212bf896160f90ea0ff0d67372.png
    Figure 8.E.9

    3. \(\left(x-\frac{1}{2}\right)^{2}+\left(y-\frac{3}{2}\right)^{2}=4\);

    65391be5d00c3fc2349d44049ee02608.png
    Figure 8.E.10

    5. \(\left(x-\frac{5}{2}\right)^{2}+\left(y+\frac{1}{2}\right)^{2}=7\);

    65032a7c2f7fe05b8648004ea1d0fe52.png
    Figure 8.E.11

    Exercise \(\PageIndex{10}\)

    Given the equation of an ellipse in standard form, determine its center, orientation, major radius, and minor radius.

    1. \(\frac{(x+12)^{2}}{16}+\frac{(y-10)^{2}}{4}=1\)
    2. \(\frac{(x+3)^{2}}{3}+\frac{y^{2}}{25}=1\)
    3. \(x^{2}+\frac{(y-5)^{2}}{12}=1\)
    4. \(\frac{(x-8)^{2}}{5}+\frac{(y+8)}{18}=1\)
    Answer

    1. Center: \((−12, 10)\); orientation: horizontal; major radius: \(4\) units; minor radius: \(2\) units

    3. Center: \((0, 5)\); orientation: vertical; major radius: \(2\sqrt{3}\) units; minor radius: \(1\) unit

    Exercise \(\PageIndex{11}\)

    Determine the standard form for the equation of the ellipse given the following information.

    1. Center \((0,-4)\) with \(a=3\) and \(b=4\)
    2. Center \((3,8)\) with \(a=1\) and \(b=\sqrt{7}\)
    3. Center \((0,0)\) with \(a=5\) and \(b=\sqrt{2}\)
    4. Center \((-10,-30)\) with \(a=10\) and \(b=1\)
    Answer

    1. \(\frac{x^{2}}{9}+\frac{(y+4)^{2}}{16}=1\)

    3. \(\frac{x^{2}}{25}+\frac{y^{2}}{2}=1\)

    Exercise \(\PageIndex{12}\)

    Find the \(x\)- and \(y\)-intercepts.

    1. \(\frac{(x+2)^{2}}{4}+\frac{y^{2}}{9}=1\)
    2. \(\frac{(x-1)^{2}}{2}+\frac{(y+1)^{2}}{3}=1\)
    3. \(5 x^{2}+2 y^{2}=20\)
    4. \(5(x-3)^{2}+6 y^{2}=120\)
    Answer

    1. \(x\) -intercepts: \((-4,0),(0,0) ; y\) -intercepts: \((0,0)\)

    3. \(x\) -intercepts: \((\pm 2,0) ; y\) -intercepts: \((0, \pm \sqrt{10})\)

    Exercise \(\PageIndex{13}\)

    Graph.

    1. \(\frac{(x-10)^{2}}{25}+\frac{(y+5)^{2}}{4}=1\)
    2. \(\frac{(x+6)^{2}}{9}+\frac{(y-8)^{2}}{36}=1\)
    3. \(\frac{\left(x-\frac{3}{2}\right)^{2}}{4}+\left(y-\frac{7}{2}\right)^{2}=1\)
    4. \(\left(x-\frac{2}{3}\right)^{2}+\frac{y^{2}}{4}=1\)
    5. \(\frac{x^{2}}{2}+\frac{y^{2}}{5}=1\)
    6. \(\frac{(x+2)^{2}}{8}+\frac{(y-3)^{2}}{12}=1\)
    Answer

    1.

    83f37f422200edabac62caf1dea0acb2.png
    Figure 8.E.12

    3.

    11ef8bcbd1481207ef0814a748f92fa3.png
    Figure 8.E.13

    5.

    debe7b837ebba6f31b5bb7e584dcb6da.png
    Figure 8.E.14

    Exercise \(\PageIndex{14}\)

    Rewrite in standard form and graph.

    1. \(4 x^{2}+9 y^{2}-8 x+90 y+193=0\)
    2. \(9 x^{2}+4 y^{2}+108 x-80 y+580=0\)
    3. \(x^{2}+9 y^{2}+6 x+108 y+324=0\)
    4. \(25 x^{2}+y^{2}-350 x-8 y+1,216=0\)
    5. \(8 x^{2}+12 y^{2}-16 x-36 y-13=0\)
    6. \(10 x^{2}+2 y^{2}-50 x+14 y+7=0\)
    Answer

    1. \(\frac{(x-1)^{2}}{9}+\frac{(y+5)^{2}}{4}=1\);

    108b4be908f56741baa53d6da6cd4ed1.png
    Figure 8.E.15

    3. \(\frac{(x+3)^{2}}{9}+(y+6)^{2}=1\);

    085c47b5ec86f9608e116005b62ab6b1.png
    Figure 8.E.16

    5. \(\frac{(x-1)^{2}}{6}+\frac{\left(y-\frac{3}{2}\right)^{2}}{4}=1\);

    d746ff18841b1d80c17d15962c850949.png
    Figure 8.E.17

    Exercise \(\PageIndex{15}\)

    Given the equation of a hyperbola in standard form, determine its center, which way the graph opens, and the vertices.

    1. \(\frac{(x-10)^{2}}{4}-\frac{(y+5)^{2}}{16}=1\)
    2. \(\frac{(x+7)^{2}}{2}-\frac{(y-8)^{2}}{8}=1\)
    3. \(\frac{(y-20)^{2}}{3}-(x-15)^{2}=1\)
    4. \(3 y^{2}-12(x-1)^{2}=36\)
    Answer

    1. Center: \((10,-5)\); opens left and right; vertices: \((8,-5),(12,-5)\)

    3. Center: \((15,20)\); opens upward and downward; vertices: \((15,20-\sqrt{3}),(15,20+\sqrt{3})\)

    Exercise \(\PageIndex{16}\)

    Determine the standard form for the equation of the hyperbola.

    1. Center \((-25,10), a=3, b=\sqrt{5},\) opens up and down.
    2. Center \((9,-12), a=5 \sqrt{3}, b=7,\) opens left and right.
    3. Center \((-4,0), a=1, b=6,\) opens left and right.
    4. Center \((-2,-3), a=10 \sqrt{2}, b=2 \sqrt{3},\) opens up and down.
    Answer

    1. \(\frac{(y-10)^{2}}{5}-\frac{(x+25)^{2}}{9}=1\)

    3. \((x+4)^{2}-\frac{y^{2}}{36}=1\)

    Exercise \(\PageIndex{17}\)

    Find the \(x\)- and \(y\)-intercepts.

    1. \(\frac{(x-1)^{2}}{4}-\frac{(y+3)^{2}}{9}=1\)
    2. \(\frac{(x+4)^{2}}{8}-\frac{(y-2)^{2}}{12}=1\)
    3. \(4(y-2)^{2}-x^{2}=16\)
    4. \(6(y+1)^{2}-3(x-1)^{2}=18\)
    Answer

    1. \(x\) -intercepts: \((1 \pm 2 \sqrt{2}, 0) ; y\) -intercepts: none

    3. \(x\) -intercepts: \((0,0) ; y\) -intercepts: \((0,0),(0,4)\)

    Exercise \(\PageIndex{18}\)

    Graph.

    1. \(\frac{(x-10)^{2}}{25}-\frac{(y+5)^{2}}{100}=1\)
    2. \(\frac{(x-4)^{2}}{4}-\frac{(y-8)^{2}}{16}=1\)
    3. \(\frac{(y-3)^{2}}{9}-\frac{(x-6)^{2}}{81}=1\)
    4. \(\frac{(y+1)^{2}}{4}-\frac{(x+1)^{2}}{25}=1\)
    5. \(\frac{y^{2}}{27}-\frac{(x-3)^{2}}{9}=1\)
    6. \(\frac{x^{2}}{2}-\frac{y^{2}}{3}=1\)
    Answer

    1.

    22e21c894d4aba25c953a397fae8fd6b.png
    Figure 8.E.18

    3.

    ae04dc1c539c3c98b806a7a4cbc57b94.png
    Figure 8.E.19

    5.

    d9c5e337e60c49caffe87fde345d8119.png
    Figure 8.E.20

    Exercise \(\PageIndex{19}\)

    Rewrite in standard form and graph.

    1. \(4 x^{2}-9 y^{2}-8 x-90 y-257=0\)
    2. \(9 x^{2}-y^{2}-108 x+16 y+224=0\)
    3. \(25 y^{2}-2 x^{2}-100 y+50=0\)
    4. \(3 y^{2}-x^{2}-2 x-10=0\)
    5. \(8 y^{2}-12 x^{2}+24 y-12 x-33=0\)
    6. \(4 y^{2}-4 x^{2}-16 y-28 x-37=0\)
    Answer

    1. \(\frac{(x-1)^{2}}{9}-\frac{(y+5)^{2}}{4}=1\);

    b562e3f2b31f31e043007fa34460cd11.png
    Figure 8.E.21

    3. \(\frac{(y-2)^{2}}{2}-\frac{x^{2}}{25}=1\);

    014249924a539a2b96566c7ccee865f1.png
    Figure 8.E.22

    5. \(\frac{\left(y+\frac{3}{2}\right)^{2}}{6}-\frac{\left(x+\frac{1}{2}\right)^{2}}{4}=1\)

    bba39f3a81a52414e6234cb8fbba0816.png
    Figure 8.E.23

    Exercise \(\PageIndex{20}\)

    Identify the conic sections and rewrite in standard form.

    1. \(x^{2}+y^{2}-2 x-8 y+16=0\)
    2. \(x^{2}+2 y^{2}+4 x-24 y+74=0\)
    3. \(x^{2}-y^{2}-6 x-4 y+3=0\)
    4. \(x^{2}+y-10 x+22=0\)
    5. \(x^{2}+12 y^{2}-12 x+24=0\)
    6. \(x^{2}+y^{2}+10 y+22=0\)
    7. \(4 y^{2}-20 x^{2}+16 y+20 x-9=0\)
    8. \(16 x-16 y^{2}+24 y-25=0\)
    9. \(9 x^{2}-9 y^{2}-6 x-18 y-17=0\)
    10. \(4 x^{2}+4 y^{2}+4 x-8 y+1=0\)
    Answer

    1. Circle;\((x-1)^{2}+(y-4)^{2}=1\)

    3. Hyperbola; \(\frac{(x-3)^{2}}{2}-\frac{(y+2)^{2}}{2}=1\)

    5. Ellipse; \(\frac{(x-6)^{2}}{12}+y^{2}=1\)

    7. Hyperbola; \(\frac{(y+2)^{2}}{5}-\left(x-\frac{1}{2}\right)^{2}=1\)

    9. Hyperbola; \(\left(x-\frac{1}{3}\right)^{2}-(y+1)^{2}=1\)

    Exercise \(\PageIndex{21}\)

    Given the graph, write the equation in general form.

    1.

    d4a159205e14de7af9b6c90a20b3d8c6.png
    Figure 8.E.24

    2.

    8b3405add73ee683d3abb5e71f019bfb.png
    Figure 8.E.25

    3.

    14e554c6a6c7ff2cc64f3d48fa25ac82.png
    Figure 8.E.26

    4.

    05443e5fb93d1991d3cd476acd549a2c.png
    Figure 8.E.27

    5.

    b84e54838134cb4347cc357008ca2d70.png
    Figure 8.E.28

    6.

    44368f995333d3197513a4e4a07442db.png
    Figure 8.E.29
    Answer

    1. \(x^{2}+y^{2}+18 x-6 y+9=0\)

    3. \(9 x^{2}-y^{2}+72 x-12 y+72=0\)

    5. \(9 x^{2}+64 y^{2}+54 x-495=0\)

    Exercise \(\PageIndex{22}\)

    Solve.

    1. \(\left\{\begin{array}{l}{x^{2}+y^{2}=8} \\ {x-y=4}\end{array}\right.\)
    2. \(\left\{\begin{array}{l}{x^{2}+y^{2}=1} \\ {x+2 y=1}\end{array}\right.\)
    3. \(\left\{\begin{array}{c}{x^{2}+3 y^{2}=4} \\ {2 x-y=1}\end{array}\right.\)
    4. \(\left\{\begin{array}{c}{2 x^{2}+y^{2}=5} \\ {x+y=3}\end{array}\right.\)
    5. \(\left\{\begin{array}{c}{3 x^{2}-2 y^{2}=1} \\ {x-y=2}\end{array}\right.\)
    6. \(\left\{\begin{array}{c}{x^{2}-3 y^{2}=10} \\ {x-2 y=1}\end{array}\right.\)
    7. \(\left\{\begin{array}{c}{2 x^{2}+y^{2}=11} \\ {4 x+y^{2}=5}\end{array}\right.\)
    8. \(\left\{\begin{array}{l}{x^{2}+4 y^{2}=1} \\ {2 x^{2}+4 y=5}\end{array}\right.\)
    9. \(\left\{\begin{array}{c}{5 x^{2}-y^{2}=10} \\ {x^{2}+y=2}\end{array}\right.\)
    10. \(\left\{\begin{array}{l}{2 x^{2}+y^{2}=1} \\ {2 x-4 y^{2}=-3}\end{array}\right.\)
    11. \(\left\{\begin{array}{c}{x^{2}+4 y^{2}=10} \\ {x y=2}\end{array}\right.\)
    12. \(\left\{\begin{array}{l}{y+x^{2}=0} \\ {x y-8=0}\end{array}\right.\)
    13. \(\left\{\begin{array}{l}{\frac{1}{x}+\frac{1}{y}=10} \\ {\frac{1}{x}-\frac{1}{y}=6}\end{array}\right.\)
    14. \(\left\{\begin{array}{l}{\frac{1}{x}+\frac{1}{y}=1} \\ {y-x=2}\end{array}\right.\)
    15. \(\left\{\begin{array}{l}{x-2 y^{2}=3} \\ {y=\sqrt{x-4}}\end{array}\right.\)
    16. \(\left\{\begin{array}{c}{(x-1)^{2}+y^{2}=1} \\ {y-\sqrt{x}=0}\end{array}\right.\)
    Answer

    1. \((2,-2)\)

    3. \(\left(-\frac{1}{13},-\frac{15}{13}\right),(1,1)\)

    5. \((-9,-11),(1,-1)\)

    7. \((-1,-3),(-1,3)\)

    9. \((-\sqrt{2}, 0),(\sqrt{2}, 0),(-\sqrt{7},-5),(\sqrt{7},-5)\)

    11. \((\sqrt{2}, \sqrt{2}) \cdot(-\sqrt{2},-\sqrt{2}) \cdot\left(2 \sqrt{2}, \frac{\sqrt{2}}{2}\right) \cdot\left(-2 \sqrt{2},-\frac{\sqrt{2}}{2}\right)\)

    13. \(\left(\frac{1}{8}, \frac{1}{2}\right)\)

    15. \((5,1)\)

    Sample Exam

    Exercise \(\PageIndex{23}\)

    1. Given two points \((-4,-6)\) and \((2,-8)\):
      1. Calculate the distance between them.
      2. Find the midpoint between them.
    2. Determine the area of a circle whose diameter is defined by the points \((4, −3)\) and \((−1, 2)\).
    Answer

    1. (1) \(2\sqrt{10}\) units; (2) \((-1,-7)\)

    Exercise \(\PageIndex{24}\)

    Rewrite in standard form and graph. Find the vertex and all intercepts if any.

    1. \(y=-x^{2}+6 x-5\)
    2. \(x=2 y^{2}+4 y-6\)
    3. \(x=-3 y^{2}+3 y+1\)
    4. Find the equation of a circle in standard form with center \((−6, 3)\) and radius \(2 \sqrt{5}\) units.
    Answer

    1. \(y=-(x-3)^{2}+4\);

    b1fe653012b461ad6510b0ed527258db.png
    Figure 8.E.30

    3. \(x=-3\left(y-\frac{1}{2}\right)^{2}+\frac{7}{4}\);

    Figure 8.E.31

    Exercise \(\PageIndex{25}\)

    Sketch the graph of the conic section given its equation in standard form.

    1. \((x-4)^{2}+(y+1)^{2}=45\)
    2. \(\frac{(x+3)^{2}}{4}+\frac{y^{2}}{9}=1\)
    3. \(\frac{y^{2}}{3}-\frac{x^{2}}{9}=1\)
    4. \(\frac{x^{2}}{16}-(y-2)^{2}=1\)
    Answer

    1.

    b46b57dacac73c77494889eb0e36d225.png
    Figure 8.E.32

    3.

    c5f335e27f3e642b70a507398d5551f7.png
    Figure 8.E.33

    Exercise \(\PageIndex{26}\)

    Rewrite in standard form and graph.

    1. \(9 x^{2}+4 y^{2}-144 x+16 y+556=0\)
    2. \(x-y^{2}+6 y+7=0\)
    3. \(x^{2}+y^{2}+20 x-20 y+100=0\)
    4. \(4 y^{2}-x^{2}+40 y-30 x-225=0\)
    Answer

    1. \(\frac{(x-8)^{2}}{4}+\frac{(y+2)^{2}}{9}=1\);

    fd25c8f2c24828374dd4797d6bfe308a.png
    Figure 8.E.34

    3. \((x+10)^{2}+(y-10)^{2}=100\);

    218569990e6d996d7d722ab601747ac4.png
    Figure 8.E.35

    Exercise \(\PageIndex{27}\)

    Find the \(x\)- and \(y\)-intercepts.

    1. \(x=-2(y-4)^{2}+9\)
    2. \(\frac{(y-1)^{2}}{12}-(x+1)^{2}=1\)
    Answer

    1. \(x\) -intercept: \((-23,0) ; y\) -intercepts: \(\left(0, \frac{8 \pm 3 \sqrt{2}}{2}\right)\)

    Exercise \(\PageIndex{28}\)

    Solve.

    1. \(\left\{\begin{array}{l}{x+y=2} \\ {y=-x^{2}+4}\end{array}\right.\)
    2. \(\left\{\begin{array}{l}{y-x^{2}=-3} \\ {x^{2}+y^{2}=9}\end{array}\right.\)
    3. \(\left\{\begin{array}{c}{2 x-y=1} \\ {(x+1)^{2}+2 y^{2}=1}\end{array}\right.\)
    4. \(\left\{\begin{array}{c}{x^{2}+y^{2}=6} \\ {x y=3}\end{array}\right.\)
    Answer

    1. \((-1,3),(2,0)\)

    3. \(\emptyset\)

    Exercise \(\PageIndex{29}\)

    1. Find the equation of an ellipse in standard form with vertices \((−3, −5)\) and \((5, −5)\) and a minor radius \(2\) units in length.
    2. Find the equation of a hyperbola in standard form opening left and right with vertices \((\pm \sqrt{5}, 0)\) and a conjugate axis that measures \(10\) units.
    3. Given the graph of the ellipse, determine its equation in general form.
    2d8acda4dda31c5e33c1c90cfdccd37a.png
    Figure 8.E.36

    4. A rectangular deck has an area of \(80\) square feet and a perimeter that measures \(36\) feet. Find the dimensions of the deck.

    5. The diagonal of a rectangle measures \(2\sqrt{13}\) centimeters and the perimeter measures \(20\) centimeters. Find the dimensions of the rectangle.

    Answer

    1. \(\frac{(x-1)^{2}}{16}+\frac{(y+5)^{2}}{4}=1\)

    3. \(4 x^{2}+25 y^{2}-24 x-100 y+36=0\)

    5. \(6\) centimeters by \(4\) centimeters

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