How many tangents to the circle x 2 + y 2 = 3 are there which are normal to the ellipse x 2 9 + y 2 4 = 1 Options: A . &nbsp3 B . &nbsp2 C . &nbsp1 D . &nbsp0

How many tangents to the circle x2+y2=3 are there which are normal to the ellips

Question

How many tangents to the circle

x^{2} + y^{2} = 3

are there which are normal to the ellipse

\frac{x^{2}}{9} + \frac{y^{2}}{4} = 1

Options:

A . 3

B . 2

C . 1

D . 0

Answer: Option D
:
D
Equation of normal at

p (3 c o s θ, 2 s i n θ)

3 x s e c θ - 2 y c o s e c θ = 5

\frac{5}{\sqrt{9 s e c^{2} θ + 4 c o s e c^{2} θ}} = \sqrt{3} B u t m i n i m u m v a l u e o f 9 s e c^{2} θ + 4 c o s e c^{2} θ = 25 ∴ n o s u c h θ^{- 1} e x i s t s

So the number of tangents to the circle

x^{2} + y^{2} = 3

which are normal to the ellipse

\frac{x^{2}}{9} + \frac{y^{2}}{4} = 1

is 0.

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