Practice!

Sample Problem

Find the foci of the ellipse 6x2+2y2-54x-10y=0

Solution

6x2+2y2-54x-10y=0

(6x2-54x)+(2y2-10y)=0

6(x2-9x)+2(y2-5y)=0

6(x2-9x+81/4)+2(y2-5y+25/4)=6(81/4)+2(25/4)

6(x-9/2)2+2(y-5/2)2=243/2+25/2=134

The major axis is vertical (a2>b2 always and therefore 67=a2 and 67/3=b2) where h=9/2, k=5/2, a=√67, b=√67/3.

c2=a2-b2

c=√(a2-b2)=√67-67/3=√(134/3)

The foci are therefore at (9/2, 5/2+√(134/3)) and (9/2, 5/2-√(134/3))