Let the equation of the ellipse be

+ = 1

Equation of a normal at any point P ( θ ) on the above ellipse, is

(a sec θ )x – (b cosec θ ) y = a 2 e 2

Let normal is passing through a point A(h, k)

⇒ (ah sec θ – a 2 e 2 ) 2 = (bk cosec θ ) 2

⇒ a 2 h 2 sec 2 θ – 2a 3 e 2 h sec θ + a 4 e 4 = b 2 k 2 cosec 2 θ = b 2 k 2

⇒ a 2 h 2 sec 4 θ – 2a 3 e 2 h sec 3 θ + (a 4 e 4 – a 2 h 2 – b 2 k 2 ) sec 2 θ + 2a 3 e 2 h sec θ – a 4 e 4 =0 ..... (1)

If α , β , γ , δ be the roots of the above equation, then

= =

Multiplying equation (1) by cos 4 θ , it reduces to

a 4 e 4 cos 4 θ – 2a 3 e 2 h cos 3 θ – (a 4 e 4 – a 2 h 2 – b 2 k 2 ) cos 2 θ + 2a 3 e 2 h cos θ – a 2 h 2 = 0 ....(2)

Then = =

Hence, we have

( ∑ sec α )( ∑ cos α ) = · = 4

which is the desired result.

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