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  • Let a line  be tangent to the hyperbola <img alt="\mathrm{\frac{x^{2}}{16}-\frac{y^{2

Let a line \mathrm{L_{1}} be tangent to the hyperbola \mathrm{\frac{x^{2}}{16}-\frac{y^{2}}{4}=1} and let \mathrm{L_{2}} be the line passing through the origin and perpendicular to \mathrm{L_{1}}. If the locus of the point of :intersection of \mathrm{L_{1}} and\mathrm{L_{2}} is \mathrm{\left(x^{2}+y^{2}\right)^{2}=\alpha x^{2}+\beta y^{2}}, then \mathrm{\alpha+\beta} is equal to

Option: 1

12


Option: 2

-


Option: 3

-


Option: 4

-


Answers (1)

best_answer

\mathrm{ \frac{x^{2}}{16}-\frac{y^{2}}{4}=1}

\mathrm{L_{1}: \quad \frac{x \sec \theta}{4}-\frac{y \tan \theta}{2}=1}

\begin{aligned} & \mathrm{m_{1}=\frac{1}{2 \sin \theta}} \\ &\mathrm{L_{2}: \quad y=-2 \sin \theta x} \end{aligned}

passes through \mathrm{(h, k)} \\

\begin{aligned} &\mathrm{k=-2(\sin \theta) h^{h}}\\ &\mathrm{\sin \theta=\frac{-k}{2 h}}\\ &\mathrm{from L_{1}}:\\ &\mathrm{\frac{h}{4} \frac{2 h}{\sqrt{4 k^{2}+k^{2}}}-\frac{k}{2}\left(\frac{-k}{\sqrt{4 k^{2}-k^{2}}}\right)=1} \\ &\mathrm{\left(x^{2}+y^{2}\right)^{2}=16 x^{2}-4 y^{2} }\\ &\mathrm{\therefore \quad \alpha=16, \quad \beta=-4 }\\ &\mathrm{\therefore \alpha+\beta=12 } \end{aligned}

 

Posted by

Ritika Kankaria

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