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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`

Text Solution

Verified by Experts

Let `P( a cos theta, b sin theta)` be a point on the ellipse `(x^(2))/(a^(2))+(y^(2))/(b^(2))=1`
Then the quation of the normal at P is `ax sec theta-"by cosec" theta=a^(2)-b^(2)`
It meets the axe at
`G((a^(2)-b^(2))/(a)cos, theta,0)and g (0,-(a^(2)-b^(2))/(b) sin theta)`
`:. PG^(2)=(acos theta-(a^(2)-b^(2))/(a)costheta)^(2)+b^(2)sin^(2)theta=(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)`
`:. PG ,Pg=b^(2):a^(2)`
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