Practice Parabola (Focus-Directrix Definition) in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

A parabola is the set of all points equidistant from a fixed point (the focus) and a fixed line (the directrix).

Every point on a parabola is exactly the same distance from the focus as it is from the directrix line. This geometric property is why satellite dishes and flashlight reflectors are parabolic—signals from the focus reflect off the curve in parallel lines.

Showing a random 20 of 50 problems.

Example 1

medium
Find the equation of the parabola with focus (0,5) and directrix y=−5.

Example 2

easy
For y2=12x, find p.

Example 3

medium
Find the focus and directrix of y=18x2.

Example 4

medium
Find the equation of the parabola with focus (0,−4) and directrix y=4.

Example 5

medium
Find the focus and directrix of the parabola y=18x2.

Example 6

easy
For x2=8y, give the directrix.

Example 7

medium
A parabola has vertex (3,1) and directrix y=4. Find its equation.

Example 8

medium
Find p for the parabola (x−1)2=16(y+2).

Example 9

medium
Find the focus and directrix of (x−1)2=12(y+2).

Example 10

easy
True or false: the vertex of a parabola is the midpoint of the focus and the foot of the perpendicular from the focus to the directrix.

Example 11

medium
A parabolic mirror has its vertex at the origin and a depth of 4 cm at a width of 8 cm. Where should a bulb be placed for parallel reflected rays?

Example 12

medium
For the parabola y2=16x, find the endpoints of the latus rectum.

Example 13

easy
A parabola has vertex at the origin and focus (0,2). Where is the directrix?

Example 14

challenge
The latus rectum (focal chord through focus, perpendicular to axis) of x2=4py has length ∣4p∣. Find it for x2=8y.

Example 15

hard
A satellite dish has a parabolic cross-section. The dish is 1.2 m across at the rim and 0.2 m deep at the center. Where (in meters from the vertex along the axis) should the receiver go?

Example 16

easy
For x2=8y, find p.

Example 17

medium
Find p for y=14x2.

Example 18

medium
Find the equation of a parabola with vertex (2,1), focus (2,4).

Example 19

challenge
A chord of the parabola y2=4x passes through the focus and has length 9. Find the equation of the chord (assume it is not the latus rectum).

Example 20

hard
A parabola has equation y=2x2−8x+7. Find its vertex, focus, and directrix.
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