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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 $$(-3 \sqrt{3}, \sqrt{6})$$

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 $$(0,-4 \sqrt{2})$$

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. $$x=5 y^{2}-5 y+2$$

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. $$x=-y^{2}-5 y+2$$

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

Figure 8.E.1

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

Figure 8.E.2

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

Figure 8.E.3

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

Figure 8.E.4

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

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. $$\left(x-\frac{3}{8}\right)^{2}+\left(y+\frac{5}{2}\right)^{2}=\frac{1}{2}$$

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 $$(0,-4)$$

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. $$x^{2}+y^{2}-4 x+2 y+1=0$$

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. $$(x+1)^{2}+(y-1)^{2}=2$$

1.

Figure 8.E.6

3.

Figure 8.E.7

5.

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. $$x^{2}+y^{2}+12 x-8 y=0$$

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

Figure 8.E.9

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

Figure 8.E.10

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

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. $$\frac{(x-8)^{2}}{5}+\frac{(y+8)}{18}=1$$

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 $$b=1$$

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. $$5(x-3)^{2}+6 y^{2}=120$$

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. $$\frac{(x+2)^{2}}{8}+\frac{(y-3)^{2}}{12}=1$$

1.

Figure 8.E.12

3.

Figure 8.E.13

5.

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. $$10 x^{2}+2 y^{2}-50 x+14 y+7=0$$

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

Figure 8.E.15

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

Figure 8.E.16

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

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. $$3 y^{2}-12(x-1)^{2}=36$$

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.

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. $$6(y+1)^{2}-3(x-1)^{2}=18$$

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. $$\frac{x^{2}}{2}-\frac{y^{2}}{3}=1$$

1.

Figure 8.E.18

3.

Figure 8.E.19

5.

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. $$4 y^{2}-4 x^{2}-16 y-28 x-37=0$$

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

Figure 8.E.21

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

Figure 8.E.22

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

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. $$4 x^{2}+4 y^{2}+4 x-8 y+1=0$$

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.

Figure 8.E.24

2.

Figure 8.E.25

3.

Figure 8.E.26

4.

Figure 8.E.27

5.

Figure 8.E.28

6.

Figure 8.E.29

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. $$\left\{\begin{array}{c}{(x-1)^{2}+y^{2}=1} \\ {y-\sqrt{x}=0}\end{array}\right.$$

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 $$(−1, 2)$$.

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.

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

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. $$\frac{x^{2}}{16}-(y-2)^{2}=1$$

1.

Figure 8.E.32

3.

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. $$4 y^{2}-x^{2}+40 y-30 x-225=0$$

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

Figure 8.E.34

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

Figure 8.E.35

Exercise $$\PageIndex{27}$$

Find the $$x$$- and $$y$$-intercepts.

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

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. $$\left\{\begin{array}{c}{x^{2}+y^{2}=6} \\ {x y=3}\end{array}\right.$$

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.

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.

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