IXL - Find the foci of a hyperbola (Algebra 2 practice)

What are the foci of the hyperbola

–

and

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Learn with an example

question

What are the foci of the hyperbola

–

and

key idea

The equation of a hyperbola that has a horizontal orientation in standard form is:

(

–

)

–

(

–

)

The center is

(

)

the vertices are

(

)

and the co-vertices are

(

)

Note that the vertices are on the hyperbola, but the co-vertices are not.

The transverse axis is the line segment with endpoints at the vertices and has length

The conjugate axis is the line segment with endpoints at the co-vertices and has length

(

)

(

–

)

(

)

(

–

)

(

)

The distance from the center to each focus is

where

The foci lie on the same line as the transverse axis. If the hyperbola has a horizontal orientation, the coordinates of the foci are

(

)

solution

To find the foci, first find the center

(

)

Then, calculate the distance from the center to the foci, which is

Find the center.

Use the equation in standard form to identify the center.

–

(

–

)

–

(

–

)

The center is

(

)

which is the origin.

Calculate

Start by identifying

and

–

is 36 and

is 64. Calculate

then take the square root.

100

Since

is a distance, it is positive. So

100

Write the foci.

Since the hyperbola has the form

(

–

)

–

(

–

)

it has a horizontal orientation. This means you should add and subtract

from the

-coordinate

of the center to find the foci. So, the foci are

(

)

(

–

)

and

(

–

)

(

)

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Teacher tools

Work it out

Not feeling ready yet? These can help:

GG.1Find the center of a hyperbola

E.1Square roots of perfect squares

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