10.8.3: Answers to Questions

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Apr 26, 2024
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- 10.8.2: Practice Test
- 11: Sequences, Probability and Counting Theory

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10.2 Section Exercises

1. An ellipse is the set of all points in the plane the sum of whose distances from two fixed points, called the foci, is a constant.

3. This special case would be a circle.

5. It is symmetric about the $x$ -axis, $y$ -axis, and the origin.

7. yes; $\dfrac{x^2}{3^2}+\dfrac{y^2}{2^2}=1$

9. yes; $\dfrac{x^2}{\left(\dfrac{1}{2}\right)^2}+\dfrac{y^2}{\left(\dfrac{1}{3}\right)^2}=1$

11. $\dfrac{x^2}{2^2}+\dfrac{y^2}{7^2}=1$ ; Endpoints of major axis $(0,7)$ and $(0,-7)$ . Endpoints of minor axis $(2,0)$ and $(-2,0)$ . Foci at $(0,3 \sqrt{5}),(0,-3 \sqrt{5})$ .

13. $\dfrac{x^2}{(1)^2}+\dfrac{y^2}{\left(\dfrac{1}{3}\right)^2}=1$ ; Endpoints of major axis $(1,0)$ and $(-1,0)$ . Endpoints of minor axis $\left(0, \dfrac{1}{3}\right),\left(0,-\dfrac{1}{3}\right)$ .

Foci at $\left(\dfrac{2 \sqrt{2}}{3}, 0\right),\left(-\dfrac{2 \sqrt{2}}{3}, 0\right)$ .

15. $\dfrac{(x-2)^2}{7^2}+\dfrac{(y-4)^2}{5^2}=1$ ; Endpoints of major axis $(9,4),(-5,4)$ . Endpoints of minor axis $(2,9),(2,-1)$ . Foci at $(2+2 \sqrt{6}, 4)^2,(2-2 \sqrt{6}, 4)$ .

17. $\dfrac{(x+5)^2}{2^2}+\dfrac{(y-7)^2}{3^2}=1$ ; Endpoints of major axis $(-5,10),(-5,4)$ . Endpoints of minor axis $(-3,7),(-7,7)$ . Foci at $(-5,7+\sqrt{5}),(-5,7-\sqrt{5})$ .

19. $\dfrac{(x-1)^2}{3^2}+\dfrac{(y-4)^2}{2^2}=1$ ; Endpoints of major axis $(4,4),(-2,4)$ . Endpoints of minor axis $(1,6),(1,2)$ . Foci at $(1+\sqrt[3]{5}, 4),(1-\sqrt{5}, 4)$ .

21. $\dfrac{(x-3)^2}{(3 \sqrt{2})^2}+\dfrac{(y-5)^2}{(\sqrt{2})^2}=1$ ; Endpoints of major axis $(3+3 \sqrt{2}, 5),(3-3 \sqrt{2}, 5)$ . Endpoints of minor axis $(3,5+\sqrt{2}),(3,5-\sqrt{2})$ . Foci at $(7,5),(-1,5)$ .

23. $\dfrac{(x+5)^2}{(5)^2}+\dfrac{(y-2)^2}{(2)^2}=1$ ; Endpoints of major axis $(0,2),(-10,2)$ . Endpoints of minor axis $(-5,4),(-5,0)$ . Foci at $(-5+\sqrt{21}, 2),(-5-\sqrt{21}, 2)$ .

25. $\dfrac{(x+3)^2}{(5)^2}+\dfrac{(y+4)^2}{(2)^2}=1$ ; Endpoints of major axis $(2,-4),(-8,-4)$ . Endpoints of minor axis $(-3,-2),(-3,-6)$ . Foci at $(-3+\sqrt{21},-4),(-3-\sqrt{21},-4)$ .

27. Foci $(-3,-1+\sqrt{11}),(-3,-1-\sqrt{11})$

29. Focus $(0,0)$

31. Foci $(-10,30),(-10,-30)$

33. Center $(0,0)$ , Vertices $(4,0),(-4,0),(0,3),(0,-3)$ , Foci $(\sqrt{7}, 0),(-\sqrt{7}, 0)$

35. Center $(0,0)$ , Vertices $\left(\dfrac{1}{9}, 0\right),\left(-\dfrac{1}{9}, 0\right),\left(0, \dfrac{1}{7}\right),\left(0,-\dfrac{1}{7}\right)$ , Foci $\left(0, \dfrac{4 \sqrt{2}}{63}\right),\left(0,-\dfrac{4 \sqrt{2}}{63}\right)$