Find the Equation of a Shifted Conic Find an equation for th

Find the Equation of a Shifted Conic. Find an equation for the conic section with the given properties.

The eclipse with foci F₁(3,-4) and F₂(3,4) and x-intercepts 0 and 6

Solution

Focii are (3,-4) and (3,4)

This ellipse is of the form

(x-h)²/b² + (y-k)²/a²=1

Where focii = (h, k+-c)

Therefore h=3   

k+c=4,k-c=-4

From these two equations we get

k=0   and   c=4

On plugging these values in the above equation we get

(x-3)²/b² +   (y-0)²/a²=1

And x intercept is 0 and 6

Therefore on plugging y=0 and x=0 we get

(0-3)²/b² +   (0-0)²/a²=1

b²=3²=9

c²=16

c²=a²-b²

16=a²-9

a²=25

Hence the required equation is

(x-3)²/9 + (y-0)²/25 =1