Question 1

Which of the following best describes the set of all points that form an ellipse?

Accepted Answer

The set of all points in a plane, the sum of whose distances from two fixed points is constant.

Question 2

Given the ellipse $$\frac{x^2$}{9} + \frac{y^2$}{4} = 1$$, what are the coordinates of its foci?

Accepted Answer

$$(\pm \sqrt{5}, 0)$$

Question 3

What is the standard form of the equation for an ellipse centered at $$(h, k)$$ with a horizontal major axis?

Accepted Answer

$$\frac{(x-h)^2$}{a^2$} + \frac{(y-k)^2$}{b^2$} = 1$$

Question 4

A lithotripter is modeled by the ellipse $$\frac{x^2$}{36} + \frac{y^2$}{27} = 1. $$How far from the center should the kidney stone and the wave source be placed?

Accepted Answer

3 units

Question 5

Which of the following is the standard form of a hyperbola centered at the origin with a horizontal transverse axis?

Accepted Answer

$$\frac{x^2$}{a^2$} - \frac{y^2$}{b^2$} = 1$$

Question 6

For the hyperbola $$\frac{x^2$}{25} - \frac{y^2$}{49} = 1$$, what are the equations of its asymptotes?

Accepted Answer

$$y = \pm \frac{7}{5}x$$

Question 7

Given the ellipse $$\frac{x^2$}{7} + \frac{y^2$}{16} = 1$$, what is its domain?

Accepted Answer

$$[-\sqrt{7}, \sqrt{7}]$$

Question 8

A hyperbola has center $$(1, 2)$$, focus $$(6, 2)$$, and vertex $$(4, 2). $$What is the value of $$b^2 in $$its standard form?

Accepted Answer

$$b^2 = 16$

Question 9

Which real-world property of ellipses is used in medical devices like lithotripters?

Accepted Answer

Reflective property: a ray from one focus reflects to the other focus.

Question 10

If the equation $$4x^2 - 16x$$ + $$9y^2 + 54y + 61 = 0$ is written in standard form, what is the center of the ellipse?

Accepted Answer

$$(2, -3)$$

Question 11

For the hyperbola $$\frac{y^2$}{9} - \frac{x^2$}{4} = 1$$, what are the coordinates of the vertices?

Accepted Answer

$$(0, \pm 3)$$

Question 12

Which of the following is true for the foci of an ellipse with equation $$\frac{x^2$}{a^2$} + \frac{y^2$}{b^2$} = 1$$ where $$a \u003e b \u003e 0$$?

Accepted Answer

The foci are at $$(\pm c, 0)$$ where $$c^2$$ = $$a^2$$ - $$b^2.$$

Question 13

A hyperbola has the equation $$\frac{(y+2)^2$}{9} - \frac{(x+3)^2$}{4} = 1. $$What is the center of this hyperbola?

Accepted Answer

$$(-3, -2)$$

Question 14

Which of the following best describes the asymptotes of the hyperbola $$\frac{x^2$}{a^2$} - \frac{y^2$}{b^2$} = 1$$?

Accepted Answer

$$y = \pm \frac{b}{a}x$$

Question 15

If an ellipse is centered at $$(h, k)$$ and has a vertical major axis, which equation represents it?

Accepted Answer

$$\frac{(x-h)^2$}{b^2$} + \frac{(y-k)^2$}{a^2$} = 1$$

Question 16

Given the ellipse $$\frac{(x-2)^2$}{9} + \frac{(y+1)^2$}{16} = 1$$, what is its range?

Accepted Answer

$$[-5, 3]$$

Question 17

For the hyperbola $$\frac{x^2$}{a^2$} - \frac{y^2$}{b^2$} = 1$$, what is the relationship between $$a$$, $$b$$, and $$c ($$the distance from the center to the foci)?

Accepted Answer

$$c^2$$ = $$a^2$$ + $$b^2$$

Ellipses and Hyperbolas: Conic Sections in Precalculus

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