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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

37-50 Finding the equation of a hyperbola Find an equation for the hyperbola that satisfies the given conditions.

Foci ( 0 , ± 3 ) , hyperbola passes through ( 1 , 4 ) .

To determine

To find:

The equation for the hyperbola.

Explanation

Given:

Foci (0,±3), hyperbola passes through (1,4)

Approach:

The general equation of a hyperbola with y-axis as transverse axis is given by

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

b2=c2a2 ...(2)

l=2c

Here, a is the transverse axis and b is the conjugate axis and c is the focus and (h,k) is the center of the hyperbola and l is the distance between the foci.

Calculation:

Since foci are (0,±3) so using the mid-point formula to find h and k

(h+h2,k+c+kc2)=(0+02,332)(h,k)=(0,0)

So, h=0, k=0

Since, the length of the conjugate axis is 2c

|3+32|=2cc=32c2=94

As, c2=a2+b2 from (2)

a2+b2=94

b2=94a2

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