   Chapter 12.4, Problem 43E ### Algebra and Trigonometry (MindTap ...

4th Edition
James Stewart + 2 others
ISBN: 9781305071742

#### Solutions

Chapter
Section ### Algebra and Trigonometry (MindTap ...

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

# SKILLS35-46 Finding the Equation of a Shifted Conic Find an equation for the conic section with the given properties.The ellipse with foci F 1 ( 1 , − 4 ) and F 2 ( 5 , − 4 ) that passes through the point ( 3 , 1 ) .

To determine

To find:

The equation of a shifted conic for ellipse with foci F1(1,4) and F2(5,4) that passes through the point (3,1).

Explanation

Approach:

Shifting graph of equations

If h and k are real numbers, then replacing x by xh or by x+h and replacing y by yk or by y+k.

Calculation:

The foci F1(1,4) and F2(5,4) is lie on the vertical axis, so equation has the form,

x2a2+y2b2=1

Replace x by xh, and y by yk.

(xh)2a2+(yk)2b2=1 ……(1)

The center is midway of the foci so the coordinates of the center will be as follows,

(h,k)=(x1+x22,y1+y22)

Substitute 1 for x1, 5 for x2, 4 for y1, and 4 for y2 in the above center formula.

(h,k)=(1+52,442)=(62,82)=(3,4)

Compare the coordinate of foci with (h±c,k). So,

h+c=53+c=5c=53=2

The ellipse passes through the point (3,1), point x=3, and y=1 must lie on ellipse.

Substitute 3 for x, 1 for y, 3 for h, and 4 for k in equation (1)

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