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If the normal at any point P on the elli...

If the normal at any point P on the ellipse cuts the major and minor axes in G and g respectively and C be the centre of the ellipse, then

A

`a^(2)(CG)^(2)+b^(2)(Cg)^(2)=(a^(2)-b^(2))^(2)`

B

`a^(2)(CG)^(2)-b^(2)(Cg)^(2) =(a^(2)-b^(2))^(2)`

C

`a^(2)(CG)^(2)-b^(2)(Cg)^(2) =(a^(2)+b^(2))^(2)`

D

None of these

Text Solution

Verified by Experts

The correct Answer is:
A
At a point `(x_(1),y_(1))` on ellipse, normal will be
`((x-x_(1))a^(2))/(x_(1)) =((y-y_(1))b^(2))/(y_(1))`
At `G, y = 0 rArr x = CG = (x_(1)(a^(2)-b^(2)))/(a^(2))`
At `g, x = 0 rArr y = Cg = (y_(1)(b^(2)-a^(2)))/(b^(2))`
Now, `(x_(1)^(2))/(a^(2)) +(y_(1)^(2))/(b^(2)) = 1 rArr a^(2) (CG)^(2) + b^(2) (Cg)^(2) = (a^(2)-b^(2))^(2)`
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