   Chapter 10.5, Problem 22E

Chapter
Section
Textbook Problem

Find the vertices, foci, and asymptotes of the hyperbola and sketch its graph.22. y2 − 16x2 = 16

To determine

To Find: The vertices, foci and asymptotes of the hyperbola for the equation y216x2=16.

Explanation

Given:

The hyperbola equation is as follows.

y216x2=16 (1).

Divide the equation (1) with the value 16.

y21616x216=1616

y216x21=1 (2).

Then, compare the equation (2) with the standard equation of hyperbola,

y2a2x2b2=1

Calculation:

Compute the center of the hyperbola using the equation:

(yk)2a2+(xh)2b2=1(y0)216+(x0)21=1

Therefore, the center of the hyperbola (h,k) is (0,0)_.

Substitute the value 16 for a2 and 1 for b2 in equation (2).

a2=16a=16a=4

b2=1b=1b=1

Compute the vertices.

vertices=(h,(k±a))

Substitute the value 0 for h,0 for k and 4 for a.

vertices=(0,(±4))

Therefore, the vertices of the hyperbola are (0,±4)_.

Compute the value of c using the equation below.

c2=a2+b2

Substitute the value 16 for a and 1 for b.

c2=(4)2+(1)2c2=16+1c=17

Compute the foci.

foci=(h,(k±c))

Substitute the value 0 for h,0 for k and 17 for c

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