Precalculus : Find the Asymptotes of a Hyperbola

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Example Question #51 : Hyperbolas

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 an equation of a line to find the equations of the asymptotes. The center of the hyperbola is .

The equations of the asymptotes can be then given by the following:

Example Question #11 : 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 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 of a line to get the equations of the asymptotes.

For the first equation, 

For the second equation,

Example Question #13 : 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 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 of alien to get the equations for the asymptotes.

The center of the hyperbola is .

The equations for the asymptotes are then:

 

Example Question #61 : Hyperbolas

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 .

To find the equations of the asymptotes, use the point-slope form of a line.

Example Question #15 : Find The Asymptotes Of A Hyperbola

Find the equations for the asymptotes of the following hyperbola:

Possible Answers:

    AND     

     AND     

Correct answer:

    AND     

Explanation:

The standard form of a hyperbola is given by

The equations for the asymptotes of a hyperbola are given by

Since our equation is already in standard form, we know h=5, k=-3, and

Plugging these vaules into the equation for the asymptote gives

 

  

  

 

So, the equations for the asymptotes are given by 

     AND     

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