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Algebra and Trigonometry (MindTap ...

4th Edition
James Stewart + 2 others
ISBN: 9781305071742

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BuyFindarrow_forward

Algebra and Trigonometry (MindTap ...

4th Edition
James Stewart + 2 others
ISBN: 9781305071742
Textbook Problem

21-28 ■ Graphing Shifted Hyperbolas An equation of hyperbola is given. (a) Find the centre, vertices, foci, and asymptotes of the hyperbola. (b) Sketch a graph showing the hyperbola and its asymptotes.

25 x 2 9 y 2 54 y = 306

To determine

(a).

To find:

The center, vertices, foci, and asymptotes of the given hyperbola.

Explanation

Given:

An equation of hyperbola

25x29y254y=306

Approach:

Completing the square of given equation, we get the equation of shifted hyperbola, then comparing the equation of given hyperbola with equation of shifted hyperbola,

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

Calculation:

Rearranging equation of hyperbola, we get

25x29y254y=30625x29(y2+6y)=30625x29(y+3)2+81=30625x29(y+3)2=225

x29(y+3)225=1

The shifted hyperbola has center, C(0,-3) and has a horizontal transverse axis.

Its graph will have similar shape as the unshifted hyperbola

x29y225=1 (Center origin)

Since a2=9 and b2=25, we get a=3, b=5, and c=a2+b2=9+25=6

To determine

(b).

To draw:

A graph showing the hyperbola and its asymptotes.

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