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Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095

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BuyFindarrow_forward

Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095
Textbook Problem

Finding the Standard Equation of a Hyperbola In Exercises 21-24, find the standard form of the equation of the hyperbola with the given characteristics.

Vertices: ( 0 , ± 8 )

Asymptotes: y = ± 2 x

To determine

To calculate: The standard form of the equation of the hyperbola whose vertices are given as, (0,±8) and asymptotes as, y=±2x.

Explanation

Given:

Vertices: (0,±8)

Asymptotes: y=±2x

Formula used:

The standard equation of a hyperbola with center (h,k), focus (h,k±c) and vertex (h,k±a) is,

(yk)2a2(xh)2b2=1

Calculation:

The vertices of a hyperbola are given as (0,±8) and the asymptotes as y=±2x.

Observing the vertices, one can notice that the hyperbola has a vertical transverse axis.

We know that for a hyperbola with vertical transverse axis, the center is (h,k), focus (h,k±c) and vertex (h,k±a).

Compare with the standard form,

h=0

Here,

2k=0k=0

And,

k+a=8a=80=8

Also, the general form of asymptotes is y=k±ab(xh)

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