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

If the normal at any point `P` on the ellipse `x^2/a^2+y^2/b^2=1` meets the axes at `G and g` respectively, then find the ratio `PG:Pg`.
(a) `a : b` (b) `a^2 : b^2` (c) `b : a` (d) `b^2 : a^2`

A

`a:b`

B

`a^(2):b^(2)`

C

`b^(2):a^(2)`

D

`b:a`

Text Solution

Verified by Experts

The correct Answer is:
C
Let P `( a cos theta, b sin theta ) ` be a point on the ellipse `(x^(2))/(a^(2))+(y^(2))/(b^(2))=1`
the equation of the normal at P is
` ax sec theta - by " cosec" theta =a^(2)-b^(2)`.
It meets axes at
`G((a^(2)-b^(2))/(a) cos theta,0) and g(0, -(a^(2)-b^(2))/(a) sin theta)`
`therefore PG^(2)=(a cos theta -(a^(2)-b^(2))/(a) cos theta)^(2)+b^(2) sin ^(2) theta`
`implies PG^(2)=(b^(2))/(a^(2))(b^(2)cos^(2) theta +a^(2)sin^(2) theta)`
` and Pg^(2) (a^(2))/(b^(2))(b^(2) cos^(2) theta +a^(2)sin ^(2)theta)`
`implies PG"Pg =b^(2):a^(2)`
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