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

If the normal at any point P of the ellipse `(x^(2))/(16)+(y^(2))/(9) =1` meets the coordinate axes at M and N respectively, then `|PM|: |PN|` equals

A

`4:3`

B

`16:9`

C

`9:16`

D

`3:4`

Text Solution

Verified by Experts

The correct Answer is:
C
The equation of the normal at `P(theta)` on the ellipse is
`4x sec theta - 3y cosec theta =7`
This meets the coordinate axes at
`M ((7)/(4) cos theta, 0), N (0,-(7)/(3) sin theta)`
`:. PM^(2) = (4-(7)/(4))^(2) cos^(2) theta + 9 sin^(2) theta`
`= (9)/(16) (9 cos^(2) theta + 16sin^(2) theta)`
`PN^(2) = 16 cos^(2) theta + (3+(7)/(3))^(2) sin theta`
`= (16)/(9) (9 cos^(2) theta + 16 sin^(2) theta)`
`:. PM^(2): PN^(2) = 9^(2): 16^(2)`
`rArr |PM| : |PN| = 9: 16`
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