   Chapter 10, Problem 22RE

Chapter
Section
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: ( ± 2 , 0 ) Asymptotes: y = ± 32 x

To determine

To calculate: The standard form of the equation of the hyperbola whose vertices are given as, (±2,0) and asymptotes as, y=±32x.

Explanation

Given:

Vertices: (±2,0)

Asymptotes: y=±32x

Formula used:

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

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

Calculation:

The vertices of a hyperbola are given as, (±2,0) and the asymptotes as, y=±32x.

On observing the vertices, it can be seen that the hyperbola has a horizontal transverse axis.

For a hyperbola with a horizontal transverse axis, the center is (h,k), focus (h±c,k) and vertex (h±a,k).

Compare with the standard form,

k=0

Here,

2h=0h=0

And,

h+a=2a=20=2

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

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