Precalculus : Find the Asymptotes of a Hyperbola

Study concepts, example questions & explanations for Precalculus

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Example Question #1 : Find The Asymptotes Of A Hyperbola

Determine the equations for the asymptotes of the following hyperbola:

 

Possible Answers:

Correct answer:

Explanation:

Let us first remember what each part of the equation for a hyperbola in standard form means:

The point (h,k) gives the center of the hyperbola. 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. For each of these standard form expressions, respectively, the equations for the asymptotes are:

So we can see that the equations are essentially the same, except that the slope is used when the hyperbola opens left and right, and its reciprocal, is used when the hyperbola opens up and down. We can see in the hyperbola for our problem that the x term appears first, so it opens left and right, which means we use the first option above for the equations of the asymptotes. Comparing the standard form equation with that given in the problem, we can determine h, k, a, and b:

Now we now all of the values required for the asymptote formula, so we plug them in and get:

So our asymptotes are described by the following two equations:

 

Example Question #2 : Find The Asymptotes Of A Hyperbola

Find the equations for the asymptotes of the following hyperbola:

Possible Answers:

Correct answer:

Explanation:

Remember that the standard form for the equation of a hyperbola is expressed in one of the following two ways:

Where a hyperbola of the first form would open left and right, and a hyperbola of the second form would open up and down. The formula for the asymptotes of either form are given below, respectively:

We can see that the equation given in the problem follows the second format of standard form, so we'll use the second equation above for finding the asymptotes of our hyperbola, which opens up and down. Looking at the equation for our hyperbola, we can see that h=-3, k= 5, a=4, and b=3. We have all the values we need, so we'll simply plug them into the formula for the asymptotes:

Example Question #3 : Find The Asymptotes Of A Hyperbola

Find the equations of the asymptotes of the hyperbola with the following equation:

Possible Answers:

Correct answer:

Explanation:

For a hyperbola with its foci on the -axis, like the one given in the equation, recall the standard form of the equation:

, where  is the center of the hyperbola.

The equations of the asymptotes for this hyperbola are given by the following equations:

For the hyperbola in question,  and .

Thus, the equations for its asymptotes are 

Example Question #4 : Find The Asymptotes Of A Hyperbola

Find the equations of the asymptotes for the hyperbola with the following equation:

Possible Answers:

Correct answer:

Explanation:

For a hyperbola with its foci on the -axis, like the one given in the equation, recall the standard form of the equation:

, where  is the center of the hyperbola.

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

Group the  terms together and  terms together.

Factor out  from the  terms and  from the  terms.

Complete the squares. Remember to add the amount amount to both sides of the equation!

Add  to both sides of the equation:

Divide both sides by .

Factor the two terms to get the standard form of the equation of a hyperbola.

The slopes of the asymptotes for this hyperbola are given by the following:

For the hyperbola in question,  and .

Thus, the slopes for its asymptotes are .

Now, use the center of the hyperbola, , to plug into the point-slope form of a line to find the equations of the asymptotes.

The first asymptote has the following equation:

The second asymptote has the following equation:

Example Question #5 : Find The Asymptotes Of A Hyperbola

Find the equations of the asymptotes for the hyperbola with the following equation:

Possible Answers:

Correct answer:

Explanation:

For a hyperbola with its foci on the -axis, like the one given in the equation, recall the standard form of the equation:

, where  is the center of the hyperbola.

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

Group the  terms together and  terms together.

Factor out  from the  terms and  from the  terms.

Complete the squares. Remember to add the amount amount to both sides of the equation!

Subtract  from both sides of the equation:

Divide both sides by .

Factor the two terms to get the standard form of the equation of a hyperbola.

The slopes of the asymptotes for this hyperbola are given by the following:

For the hyperbola in question,  and .

Thus, the slopes for its asymptotes are .

Plug in the center of the hyperbola into the point-slope form of a line to find the equations of the asymptotes. The center of the hyperbola is .

Now, simplify each equation. For the first equation,

For the second equation,

Example Question #6 : Find The Asymptotes Of A Hyperbola

Find the equations of the asymptotes for the hyperbola with the following equation:

Possible Answers:

Correct answer:

Explanation:

For a hyperbola with its foci on the -axis, like the one given in the equation, recall the standard form of the equation:

, where  is the center of the hyperbola.

The slopes of the asymptotes for this hyperbola are given by the following:

For the hyperbola in question,  and .

Thus, the slopes for its asymptotes are 

Now, plug in the coordinates for the center of the hyperbola  into the point-slope form of the line to find the equations of the asymptotes.

Example Question #7 : Find The Asymptotes Of A Hyperbola

Find the equations of the asymptotes for the hyperbola with the following equation:

Possible Answers:

Correct answer:

Explanation:

For a hyperbola with its foci on the -axis, like the one given in the equation, recall the standard form of the equation:

, where  is the center of the hyperbola.

The slopes of the asymptotes for this hyperbola are given by the following:

For the hyperbola in question,  and .

Thus, the slopes for its asymptotes are .

Now, use the point-slope form of a line in addition to the center of the hyperbola to find the equations of the asymptotes.

The center is at .

The first equation of the asymptote would be the following:

The second equation of the asymptote would be the following:

 

Example Question #8 : Find The Asymptotes Of A Hyperbola

Find the equations of the asymptotes for the hyperbola with the following equation:

Possible Answers:

Correct answer:

Explanation:

For a hyperbola with its foci on the -axis, like the one given in the equation, recall the standard form of the equation:

, where  is the center of the hyperbola.

The slopes of the asymptotes for this hyperbola are given by the following:

For the hyperbola in question,  and .

Since the center is , the equations for its asymptotes are .

Example Question #9 : Find The Asymptotes Of A Hyperbola

Find the equations of the asymptotes for the hyperbola with the following equation:

Possible Answers:

Correct answer:

Explanation:

For a hyperbola with its foci on the -axis, like the one given in the equation, recall the standard form of the equation:

, where  is the center of the hyperbola.

The slopes of the asymptotes for this hyperbola are given by the following:

For the hyperbola in question,  and .

Thus, the slopes for its asymptotes are .

Now, plug in the center of the hyperbola,  into the point-slope form of a line to find the equations of the asymptotes.

For the first asymptote, 

For the second asymptote,

Example Question #10 : Find The Asymptotes Of A Hyperbola

Find the equations of the asymptotes for the hyperbola with the following equation:

Possible Answers:

Correct answer:

Explanation:

For a hyperbola with its foci on the -axis, like the one given in the equation, recall the standard form of the equation:

, where  is the center of the hyperbola.

The slopes of this hyperbola are given by the following:

For the hyperbola in question,  and .

Thus, the slopes for its asymptotes are .

Now, plug in the center of the hyperbola into the point-slope form of the equation fo the line to get the equations of the asymptotes.

The center of the hyperbola is . The equations of the asymptotes are then:

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