what is the relation between the slopes of 3 concurrent normals of an ellipse

The normal to a curve is a line perpendicular to the tangent to curve through the point of contact.

 

As discussed above in case of tanegnts, the various ways of finding the normal are listed below:

 

A line y = mx + c is normal to the ellipse x2/a2 + y2/b2 = 1 if c2 = m2 (a2 - b2)2/ (a2 + b2m2)

 

Point form: The equation of the normal to the ellipse x2/a2 + y2/b2 = 1 at the point (x1, y1) is a2x/x1 - b2y/y1 = a2 - b2.

 

Parametric form: The equtaion of the normal to the ellipse x2/a2 + y2/b2 = 1 at the point (a cos θ, b sin θ) is ax sec θ - by cosec θ = a2 - b2.

 

ope Form: The equation of normal to ellipse x2/a2 + y2/b2 = 1 in terms of slope is given by y = mx ± m(a2-b2)/√(a2 + b2m2). The coordinates of the point of contact are (((± a2)/√(a2 + b2m2), (± b2m)/√(a2 + b2m2))