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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
BuyFindarrow_forward

Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

Find an equation for the ellipse that shares a vertex and a focus with the parabola x2 + y = 100 and that has its other focus at the origin.

To determine

To find: The equation of the ellipse that can share a vertex and a focus is to be derived. The equation should be derived using the parabolic equation x2+y=100 and other focus with the origin.

Explanation

Given:

Equation of the parabola is x2+y=100 (1)

Calculation:

Rearrange the equation (1) as below.

x2=100y

x2=(y100) (2)

Use the equation (2).

The vertices of the ellipse is (0,100) .

Another equation of the parabola in the standard form is as below.

(xh)2=4p(yk) (3)

The vertex of the parabola is given as below

(h,k)=(0,100)

Compare equation (2) and (3).

4p=1

p=14

The shared focus of the parabola is at point (h,k+p) =(0,10014)=(0,3994) .

(h,k+p)=(0,10014)

(h,k+p)=(0,3994) (4)

Refer the equation (4). Latex rectum is of the equation 2c=3994 .

c=3998 (5)

Center of the ellipse is (h,k)=(0,3998) (6)

a=1003998=8003998

a=4018 (7)

Use the below equation for ellipse.

c2=a2b2 (8)

Substitute the expressions from equation (5) and (7) in equation (8)

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