Question

Number of distinct normal lines can be drawn to ellipse x^2/169+y62/25=1 from the point (0,6)is ?

Answer 1

point (0,6)is lying outside the ellipse
equation of the ellipse at P is 
5xsec∅-3ycosec∅=16
the equation of a normal to the circle=x²+y²=25 at point Q is y=x tan∅
Q=(5cos∅,5sin∅)
P=(5cos∅,3sin∅)
we then eliminate equation (i) &( ii):x²+y²=64
x²/169+y²/25=1
the equation of the normal at point:(13cos ∅,5sin∅) is;=13x/cos∅-5y/sin∅=144
it passes through (0,6),therefore;
(15+72sin∅=0)or (sin∅=-5/24)
OR
∅=2π-sin^-1(5/24) & π+sin^-1(5/24)
hence,y axis is also one of the normals.

Answer 2

Kindly see the provided attachment for the answer