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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643

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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

Find an equation for the conic that satisfies the given conditions.

48. Hyperbola, foci (2, 0), (2, 8), asymptotes y = 3 + 1 2 x and y = 5 1 2 x

To determine

To Find: The equation for the conic using the foci (2,0) , (2,8) and asymptotes y=3+12x , and y=512x of the Hyperbola.

Explanation

Given:

The foci and asymptotes of the hyperbola are (2,0) , (2,8) and y=3+12x , y=512x .

Calculation:

The x coordinate of both the foci are same. The value of h is 2 .

Compute the value of c using the equation below:

2c=(focus)1(focus)2

Substitute 0 for (focus)1 and 8 for (focus)2 .

2c=082c=8c=±4

Therefore, the value of c is ±4 .

Compute the center of the hyperbola using the below equation.

focus=(h,(k±c))

Substitute the value ±4 for c , 2 for h and (2,0) for vertices.

(2,0)=(2,(k±c))(k4)=0k=4

Therefore, the center of the hyperbola (h,k) is (2,4) .

Refer to the value of vertices the foci. The vertices are located in the y axis.

The equation of asymptotes is as below.

y=±(ab)x

Compare the above equation, with the given equation to get the asymptotes value.

y=3+12x

Then the slope ab is 12 .

b=2a

Compute the value of a and b using the below equation.

c2=a2+b2

Substitute the value 2a for b and 4 for c

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