Question 1

Given the parabola with vertex $$(0,3)$$ and focus $$(3,3)$$, which equation correctly represents its graph?

Accepted Answer

$$(y-3)^2 = 12x$

Question 2

A parabola has its vertex at $$(1,3)$$ and opens to the right. Its focus is at $$(5,3). $$What is the equation of the parabola?

Accepted Answer

$$(y-3)^2 = 16(x-1$$)$$

Question 3

For the parabola with vertex $$(4,2)$$ and axis of symmetry along the $$y-$$axis, which equation describes a parabola opening upward?

Accepted Answer

$$(x-4)^2 = 4a(y-2$$)$$

Question 4

A parabola opens to the right and has vertex $$(-7,0)$$ and focus $$(9,0). $$What is the equation of the parabola?

Accepted Answer

$$(y-0)^2 = 32(x+7$$)$$

Question 5

Given a parabola with vertex $$(-1,-3)$$ and focus $$(-1,-4)$$, which equation represents the parabola opening downward?

Accepted Answer

$$(x+1)^2 = -4(y+3$$)$$

Question 6

An ellipse is centered at $$(0,0)$$ with vertices at $$(0,4)$$ and $$(0,-4)$$, and co-vertices at $$(2,0)$$ and $$(-2,0). $$What is its equation?

Accepted Answer

$$\frac{x^2$}{4} + \frac{y^2$}{16} = 1$$

Question 7

For the ellipse $$\frac{y^2$}{4} + \frac{x^2$}{16} = 1$$, what are the coordinates of its foci?

Accepted Answer

$$(\pm 2\sqrt{3}, 0)$$

Question 8

An ellipse is centered at $$(0,0)$$ with vertices at $$(0,6)$$ and $$(0,-6)$$, and co-vertices at $$(4,0)$$ and $$(-4,0). $$What is the value of $$c$$, the distance from the center to each focus?

Accepted Answer

$$c = \sqrt{36 - 16} = \sqrt{20}$$

Question 9

Given the ellipse $$\frac{(x-2)^2$}{4} + \frac{(y-0)^2$}{36} = 1$$, what are the coordinates of its center and lengths of its axes?

Accepted Answer

Center $$(2,0)$$, major axis $$12$$, minor axis $$4$$

Question 10

A parabola has equation $$(y-4)^2 = -4(x-0$$)$$. What is the direction in which it opens and the coordinates of its focus?

Accepted Answer

Opens left, focus $$(-1,4)$$

Question 11

For the ellipse $$\frac{(x+4)^2$}{9} + \frac{(y-4)^2$}{36} = 1$$, what are the coordinates of its vertices?

Accepted Answer

$$(-4, 10)$$ and $$(-4, -2)$$

Question 12

A parabola has equation $$(y-3)^2 = 4(x-1$$)$$. What is the length of the latus rectum?

Accepted Answer

$$4$$

Question 13

Given the ellipse $$\frac{(x-2)^2$}{4} + \frac{(y-0)^2$}{36} = 1$$, what is the equation of the latus rectum?

Accepted Answer

$$x = 2 \pm 2$$

Question 14

For the ellipse $$\frac{x^2$}{4} + \frac{y^2$}{16} = 1$$, what is the distance between the foci?

Accepted Answer

$$2c = 8$$

Question 15

A parabola has equation $$(y-3)^2 = 12x$. What is the distance from the vertex to the focus?

Accepted Answer

$$3$$

Question 16

Given the ellipse $$\frac{(x-2)^2$}{4} + \frac{(y-0)^2$}{36} = 1$$, what is the value of $$c ($$distance from center to focus)?

Accepted Answer

$$c = \sqrt{36 - 4} = \sqrt{32}$$

Question 17

For the parabola $$(y-3)^2 = 12x$, what are the coordinates of the directrix?

Accepted Answer

$$x = -3$$

Question 18

An ellipse has equation $$\frac{(x-2)^2$}{4} + \frac{(y-0)^2$}{36} = 1. $$What is the length of the major axis?

Accepted Answer

$$12$$

Conic Sections: Parabolas and Ellipses – Study Notes for Precalculus

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