Practice Hyperbola in Math

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

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Quick Recap

The set of all points in a plane where the absolute difference of the distances to two fixed points (foci) is constant. The curve has two separate branches and asymptotes.

While an ellipse keeps the SUM of distances to foci constant, a hyperbola keeps the DIFFERENCE constant. This creates two separate curves that open away from each other, each curving toward (but never reaching) a pair of asymptotic lines.

Showing a random 20 of 50 problems.

Example 1

hard
Find the equation of a hyperbola with eccentricity 2\sqrt{2} and vertex at (3,0)(3, 0) centered at origin.

Example 2

medium
Convert 4x2−9y2=364x^2 - 9y^2 = 36 to standard form.

Example 3

easy
Find the center of (x+4)29−(y−1)216=1\frac{(x+4)^2}{9} - \frac{(y-1)^2}{16} = 1.

Example 4

hard
Find the eccentricity of x216−y220=1\frac{x^2}{16} - \frac{y^2}{20} = 1.

Example 5

hard
A hyperbola has center at the origin, a vertex at (0,6)(0, 6), and passes through (8,10)(8, 10). Find its equation.

Example 6

hard
Identify and convert 4x2−y2−16x−4y+16=04x^2 - y^2 - 16x - 4y + 16 = 0 to standard form.

Example 7

medium
Find the center and foci of (x−1)29−(y+2)216=1\frac{(x-1)^2}{9}-\frac{(y+2)^2}{16}=1.

Example 8

easy
Find the vertices of x236−y249=1\frac{x^2}{36} - \frac{y^2}{49} = 1.

Example 9

medium
Find the asymptotes of 9x2−4y2=369x^2 - 4y^2 = 36.

Example 10

challenge
Find the points of intersection between x216−y29=1\frac{x^2}{16} - \frac{y^2}{9} = 1 and the line y=xy = x.

Example 11

challenge
Find the eccentricity of a hyperbola whose asymptotes are y=±xy=\pm x.

Example 12

medium
Find the foci of the hyperbola x225−y2144=1\frac{x^2}{25} - \frac{y^2}{144} = 1.

Example 13

easy
For a hyperbola, the relation between aa, bb, and cc is ____.

Example 14

easy
Find the vertices of x225−y24=1\frac{x^2}{25}-\frac{y^2}{4}=1.

Example 15

hard
Find the equation of the hyperbola with foci (±10,0)(\pm 10, 0) and asymptotes y=±43xy = \pm \frac{4}{3} x.

Example 16

medium
Find the eccentricity of x216−y29=1\frac{x^2}{16}-\frac{y^2}{9}=1.

Example 17

medium
Find the foci of x216−y29=1\frac{x^2}{16}-\frac{y^2}{9}=1.

Example 18

challenge
Write the equation of a hyperbola with asymptotes y=±23xy=\pm\frac{2}{3}x and a vertex at (3,0)(3,0).

Example 19

medium
Find the foci of x29−y240=1\frac{x^2}{9} - \frac{y^2}{40} = 1.

Example 20

medium
Find the asymptotes of y216−x29=1\frac{y^2}{16}-\frac{x^2}{9}=1.
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