Consider an ellipse whose focus is at (ae,0) and directrix along (x=ae).
Find the equation of the ellipse if
b2=a2(1−e2)
x2a2+y2b2=1
As per definition of ellipse, Let the point be S(h, k), focus be (p,q) and directrix be lm+my+n=0
⇒Distance between s &(p,q)Distance from lx+my+n=0=e(h−p)2+(k−q)2=e2(hl+mk+nl√l2+m2)2⇒(l2+m2)[(h−p)2+(k−q)2]=e2(lh+mk+n)2Here l=1, m=0, n=−ae, p=ae, q=0[(h−ae)2+(k−0)2]=e2(lh+(0)k+(−ae))2h2+k2=(eh)2+a2(1−e2)h2(a2)+k2(a2)(1−e2)=1will set b2=a2(1−e2)∴ x2a2+y2b2=1