If the pair of lines given by $(x \cos \alpha+y \sin \alpha)^2=\left(x^2+y^2\right) \sin ^2 \alpha$ are perpendicular to each other, then $\alpha$ is

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Aicharya · Accepted Answer

$\begin{array}{ll} & (x \cos \alpha+y \sin \alpha)^2=\left(x^2+y^2ight) \sin ^2 \alpha \ 	herefore \quad & x^2 \cos ^2 \alpha+y^2 \sin ^2 \alpha+2 x y \sin \alpha \cos \alpha \ & =x^2 \sin ^2 \alpha+y^2 \sin ^2 \alpha \ 	herefore \quad & x^2\left(\cos ^2 \alpha-\sin ^2 \alphaight)+2 x y \sin \alpha \cos \alpha=0 \end{array}$ This represents a pair of straight lines where $\mathrm{a}=\cos ^2 \alpha-\sin ^2 \alpha, \mathrm{h}=\sin \alpha \cos \alpha$ and $\mathrm{b}=0$ As lines are perpendicular, we get $\begin{aligned} & a+b=0 \ & 	herefore \quad \cos ^2 \alpha=\sin ^2 \alpha \Rightarrow \alpha=\frac{\pi}{4} \end{aligned}$

If the pair of lines given by $(x \cos \alpha+y \sin \alpha)^2=\left(x^2+y^2\right) \sin ^2 \alpha$ are…

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