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  • If L be the length of common tangent to the ellipse <img alt="\mathrm{\frac{x^2}{25}+\frac{y^2}{4}=1}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7B%5Cfrac%7Bx%5E2%7D%7B25%7D&am

If L be the length of common tangent to the ellipse \mathrm{\frac{x^2}{25}+\frac{y^2}{4}=1} and the circle \mathrm{x^2+y^2=16} intercepted by the coordinate axis, then \mathrm{\frac{\sqrt{3} L}{2}} is

Option: 1

7


Option: 2

6


Option: 3

8


Option: 4

9


Answers (1)

best_answer

\mathrm{\text { The equation of the tangent at }(5 \cos \theta, 2 \sin \theta) \text { is }}

\mathrm{\frac{x}{5} \cos \theta+\frac{y}{2} \sin \theta=1 \text {. }}

\mathrm{\text { If it is a tangent to the circle, then } \frac{1}{\sqrt{\frac{\cos ^2 \theta}{25}+\frac{\sin ^2 \theta}{4}}}=4}

\mathrm{\Rightarrow \cos \theta=\frac{10}{4 \sqrt{7}}, \sin \theta=\frac{\sqrt{3}}{2 \sqrt{7}}}

Let A and B be the points where the tangent meets the coordinate axis, then

\mathrm{A\left(\frac{5}{\cos \theta}, 0\right), B\left(0, \frac{2}{\sin \theta}\right) ; L=\sqrt{\frac{25}{\cos ^2 \theta}+\frac{4}{\sin ^2 \theta}}=\frac{14}{\sqrt{3}}}

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Divya Prakash Singh

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