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How many tangents to the circle x^2 + y...

How many tangents to the circle `x^2 + y^2 = 3` are normal tothe ellipse `x^2/9+y^2/4=1?`

A

3

B

2

C

1

D

0

Text Solution

Verified by Experts

The correct Answer is:
D
Equation of normal to ellipse `(x^(2))/(9) + (y^(2))/(4) =1` at `P(3 cos theta, 2 sin theta)` is
`3x sec theta -2y cos theta =5`
If it is tangent to circle `x^(2) + y^(2) =3`, then
`rArr (5)/(sqrt(9sec^(2) theta+4 cosec^(2)theta)) =sqrt(3)`
`9 sec^(2) theta + 4 cosec^(2) theta = 9 = 4+ 9 tan^(2) theta + 4 cot^(2) theta = 25 +(3 tan theta -2 cot theta)^(2)`
`:. (9 sec^(2) theta + 4 cosec^(2) theta)_(min) =25`
`:.` no such `theta` exists.
Hence no tangents to circle which is normal to ellipse.
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