if the line lx+my+n=0 is a normal to the ellipse x2/a2 + y2/b2 =1 then prove that a2/l2 +b2/m2 = - Maths - Conic Sections

Question

if the line lx+my+n=0 is a normal to the ellipse
x2/a2 + y2/b2 =1 then
prove that a2/l2 +b2
​/m2 =
(a2-b2)2/n2????

Rahul Raj · Accepted Answer

dear student

normals of slope m to the ellipse are given by=mx±m(a2-b2)a2+b2m2so for y=mx+cc=±m(a2-b2)a2+b2m2c2=m2(a2-b2)2a2+b2m2for lx+my+n=0y=-lmx-nmc=-n/mslope=-l/mc2=m2(a2-b2)2a2+b2m2n2m2=l2m2(a2-b2)2a2+b2l2m2a2m2+b2l2l2m2=(a2-b2)2n2a2l2+b2l2m2=(a2-b2)2n2

regards

if the line lx+my+n=0 is a normal to the ellipse x2/a2 + y2/b2 =1 then prove that a2/l2 +b2 ​/m2 = (a2-b2)2/n2????

if the line lx+my+n=0 is a normal to the ellipse x²/a² + y²/b² =1 then prove that a²/l² +b² /m² = (a²-b²)²/n²????