Precalculus : Find the Foci of a Hyperbola

Study concepts, example questions & explanations for Precalculus

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Example Questions

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Example Question #15 : Hyperbolas

Find the foci of the hyperbola with the following equation:

Possible Answers:

Correct answer:

Explanation:

Recall that the standard formula of a hyperbola can come in two forms:

 and

, where the centers of both hyperbolas are .

When the term with  is first, that means the foci will lie on a horizontal transverse axis.

When the term with  is first, that means the foci will lie on a vertical transverse axis.

To find the foci, we use the following:

For a hyperbola with a horizontal transverse access, the foci will be located at  and .

For a hyperbola with a vertical transverse access, the foci will be located at  and .

For the given hypebola in the question, the transverse axis is vertical and its center is located at .

Next, find .

The foci are then located at  and .

Example Question #16 : Hyperbolas

Find the foci for the hyperbola with the following equation:

Possible Answers:

Correct answer:

Explanation:

Recall that the standard formula of a hyperbola can come in two forms:

 and

, where the centers of both hyperbolas are .

When the term with  is first, that means the foci will lie on a horizontal transverse axis.

When the term with  is first, that means the foci will lie on a vertical transverse axis.

To find the foci, we use the following:

For a hyperbola with a horizontal transverse access, the foci will be located at  and .

For a hyperbola with a vertical transverse access, the foci will be located at  and .

For the given hypebola in the question, the transverse axis is vertical and its center is located at .

Next, find .

The foci are then located at  and .

Example Question #121 : Conic Sections

Which point is one of the foci of the hyperbola ?

Possible Answers:

Correct answer:

Explanation:

To find the foci of a hyperbola, first determine a and b, and then use the relationship

In this case, the major axis is horizontal since x comes first, so and .

Solve for c: add 9 to both sides 

take the square root

Since the center is and the major axis is the horizontal one, our foci are . The only choice that works is

.

Example Question #11 : Hyperbolas

Determine the length of the foci for the following hyperbola equation:

Possible Answers:

Correct answer:

Explanation:

To solve, simply use the follow equation where c is the length of the foci.

In this particular case,

Thus,

Example Question #21 : Hyperbolas

Find the foci of the hyperbola with the following equation:

Possible Answers:

 and 

 and 

 and 

 and 

 and 

Correct answer:

 and 

Explanation:

The standard form of the equation for a hyperbola is given by

The foci are located at (h+c, k) and (h-c, k), where c is found by using the formula

Since our equation is already in standard form, you can see that

Plugging into the formula

So the foci are found at

 AND

 

 

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