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  • If the radius of the largest circle with centre  inscribed in the ellipse <img alt="x^{2}+4 y^{2}=36"

If the radius of the largest circle with centre (2,0) inscribed in the ellipse x^{2}+4 y^{2}=36 is r,then 12 r^{2} is equal to

Option: 1

69


Option: 2

72


Option: 3

115


Option: 4

92


Answers (1)

best_answer

\mathrm{C}(2,0)
Ellipse \: \: x^{2}+4 y^{2}=36
\frac{x^{2}}{36}+\frac{y^{2}}{9}=1

Equation of Normal at P(6 \cos \theta, 3 \sin \theta)$ is $(6 \sec \theta) x-(3 \operatorname{cosec} \theta) y=27
It passes through \left ( 2,0 \right )

\Rightarrow \sec \theta=\frac{27}{12}=\frac{9}{4}
\cos \theta \frac{4}{9}, \sin \theta=\frac{\sqrt{65}}{9}

\mathrm{P}\left(\frac{8}{3}, \frac{\sqrt{65}}{3}\right)
\frac{\gamma}{\mathrm{p}\left(\frac{8}{3}, \frac{\sqrt{65}}{3}\right) \mathrm{c}(2,0)}
\gamma=\sqrt{\left(\frac{8}{3}-2\right)^{2}+\left(\frac{\sqrt{65}}{3}\right)^{2}}=\frac{\sqrt{69}}{3}
Value \: of \: \: 12 \gamma^{2}=\left(\frac{\sqrt{69}}{3}\right)^{2} \times 12
\Rightarrow \frac{12 \times 69}{9}=92

 

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Nehul

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