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Let L be common tangent line to the curves 4x^2 +9y^2=36 \text{ and } (2x)^2+(2y)^2=31. Then the square of the slope of the line L is _________.
Option: 1 3
Option: 2 4
Option: 3 5
Option: 4 6

Answers (1)

best_answer

\\\text { Given curves are } \frac{\mathrm{x}^{2}}{9}+\frac{\mathrm{y}^{2}}{4}=1\;,\;x^{2}+y^{2}=\frac{31}{4}

Let slope of common tangent be m

\begin{aligned} &\text { so tangents are } y=m x \pm \sqrt{9 m^{2}+4}\\ &y=m x \pm \frac{\sqrt{31}}{2} \sqrt{1+m^{2}} \end{aligned}

\\\text { hence } 9 \mathrm{~m}^{2}+4=\frac{31}{4}\left(1+\mathrm{m}^{2}\right) \\ \Rightarrow 36 \mathrm{~m}^{2}+16=31+31 \mathrm{~m}^{2} \Rightarrow \mathrm{m}^{2}=3

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himanshu.meshram

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