Precalculus : Find the Center, Vertices, and the Endpoints of the Conjugate Axis of a Hyperbola

Study concepts, example questions & explanations for Precalculus

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Example Questions

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Example Question #1 : Find The Center, Vertices, And The Endpoints Of The Conjugate Axis Of A Hyperbola

Find the center and the vertices of the following hyperbola:

Possible Answers:

Correct answer:

Explanation:

In order to find the center and the vertices of the hyperbola given in the problem, we must examine the standard form of a hyperbola:

The point (h,k) gives the center of the hyperbola. We can see that the equation in this problem resembles the second option for standard form above, so right away we can see the center is at:

In the first option, where the x term is in front of the y term, the hyperbola opens left and right. In the second option, where the y term is in front of the x term, the hyperbola opens up and down. In either case, the distance tells how far above and below or to the left and right of the center the vertices of the hyperbola are. Our equation is in the first form, where the x term is first, so the hyperbola opens left and right, which means the vertices are a distance  to the left and right of the center. We can now calculate  by identifying it in our equation, and then go 3 units to the left and right of our center to find the following vertices:

Example Question #2 : Find The Center, Vertices, And The Endpoints Of The Conjugate Axis Of A Hyperbola

Find the center of the hyperbola with the following equation:

Possible Answers:

Correct answer:

Explanation:

Recall the standard form of the equation of a hyperbola can come in two forms:

 and 

In both cases, the center of the hyperbola is located at .

For the hyperbola in question,  and , so the center is at .

Example Question #3 : Find The Center, Vertices, And The Endpoints Of The Conjugate Axis Of A Hyperbola

Find the center of the hyperbola with the following equation:

Possible Answers:

Correct answer:

Explanation:

Recall the standard form of the equation of a hyperbola can come in two forms:

 and 

In both cases, the center of the hyperbola is located at .

For the hyperbola in question,  and , so the center is at .

Example Question #4 : Find The Center, Vertices, And The Endpoints Of The Conjugate Axis Of A Hyperbola

Find the vertices of the hyperbola with the following equation:

Possible Answers:

Correct answer:

Explanation:

Recall the standard form of the equation of a hyperbola can come in two forms:

 and 

In both cases, the center of the hyperbola is located at .

For the hyperbola with the equation , the vertices are located at .

For the hyperbola with the equation , the vertices are located at .

 

For the hyperbola in question, the center is located at  and . The vertices must be at  and .

Example Question #5 : Find The Center, Vertices, And The Endpoints Of The Conjugate Axis Of A Hyperbola

Find the vertices of a hyperbola with the following equation:

Possible Answers:

Correct answer:

Explanation:

Recall the standard form of the equation of a hyperbola can come in two forms:

 and 

In both cases, the center of the hyperbola is located at .

For the hyperbola with the equation , the vertices are located at .

For the hyperbola with the equation , the vertices are located at .

 

For the hyperbola in question, the center is located at  and . The vertices must be at  and .

Example Question #6 : Find The Center, Vertices, And The Endpoints Of The Conjugate Axis Of A Hyperbola

Find the endpoints of the conjugate vertices of the hyperbola with the following equation:

Possible Answers:

Correct answer:

Explanation:

Recall the standard form of the equation of a hyperbola can come in two forms:

 and 

In both cases, the center of the hyperbola is located at .

For the hyperbola with the equation , the endpoints of the conjugate axis are located at .

For the hyperbola with the equation , the endpoints of the conjugate axis are located at .

For the hyperbola in question, the center is located at  and . The endpoints of the conjugate axis must be at  and .

Example Question #7 : Find The Center, Vertices, And The Endpoints Of The Conjugate Axis Of A Hyperbola

Find the endpoints of the conjugate axis for the hyperbola with the following equation:

Possible Answers:

Correct answer:

Explanation:

Recall the standard form of the equation of a hyperbola can come in two forms:

 and 

In both cases, the center of the hyperbola is located at .

For the hyperbola with the equation , the endpoints of the conjugate axis are located at .

For the hyperbola with the equation , the endpoints of the conjugate axis are located at .

For the hyperbola in question, the center is located at  and . The endpoints of the conjugate axis must be at  and .

Example Question #8 : Find The Center, Vertices, And The Endpoints Of The Conjugate Axis Of A Hyperbola

Find the center of the hyperbola with the following equation:

Possible Answers:

Correct answer:

Explanation:

Recall the standard form of the equation of a hyperbola can come in two forms:

 and 

In both cases, the center of the hyperbola is located at .

Start by putting the given equation into the standard form of the equation of a hyperbola.

Factor out  from the  terms.

Now complete the square. Remember to add the same amount to both sides of the equation!

Add  to both sides of the equation.

Factor the square portion of the equation.

Divide both sides by  to get the standard form of the equation of a hyperbola..

For the hyperbola in question,  and , so the center is at .

Example Question #9 : Find The Center, Vertices, And The Endpoints Of The Conjugate Axis Of A Hyperbola

Find the center of the hyperbola with the following equation:

Possible Answers:

Correct answer:

Explanation:

Recall the standard form of the equation of a hyperbola can come in two forms:

 and 

In both cases, the center of the hyperbola is located at .

Start by putting the given equation into the standard form of the equation of a hyperbola.

Group the  terms together and the  terms together.

Factor out  from the  terms and  from the  terms.

Now complete the squares. Remember to add the same amount to both sides of the equation!

Subtract  from both sides of the equation.

Factor the square portions of the equation.

Divide both sides by  to get the standard form of the equation of a hyperbola.

For the hyperbola in question,  and , so the center is at .

Example Question #44 : Hyperbolas

Find the vertices of the hyperbola with the following equation:

Possible Answers:

Correct answer:

Explanation:

Recall the standard form of the equation of a hyperbola can come in two forms:

 and 

In both cases, the center of the hyperbola is located at .

For the hyperbola with the equation , the vertices are located at .

For the hyperbola with the equation , the vertices are located at .

 

Start by putting the given equation into the standard form of the equation of a hyperbola.

Group the  terms together and the  terms together.

Factor out  from the  terms and  from the  terms.

Now complete the squares. Remember to add the same amount to both sides of the equation!

Add  to both sides of the equation.

Factor the square portions of the equation.

Divide both sides by  to get the standard form for the equation of a hyperbola.

For the hyperbola in question, the center is located at  and . The vertices must be at  and .

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