If the angle between the lines represented by the equation $x^2+\lambda x y-y^2 an ^2 heta=0$ is $2 heta$, then the value of $\lambda$ is

Question

If the angle between the lines represented by the equation $x^2+\lambda x y-y^2 	an ^2 	heta=0$ is $2 	heta$, then the value of $\lambda$ is

Aicharya · Accepted Answer

Given equation of pair of lines is $\begin{array}{ll} & x^2+\lambda x y-y^2 	an ^2 	heta=0 \ 	herefore \quad & \mathrm{a}=1, \mathrm{~h}=\frac{\lambda}{2}, \mathrm{~b}=-	an ^2 	heta \ 	herefore \quad & 	an 2 	heta=\left|\frac{2 \sqrt{\mathrm{h}^2-\mathrm{ab}}}{\mathrm{a}+\mathrm{b}}ight| \ \Rightarrow & \frac{2 	an 	heta}{1-	an ^2 	heta}=\left|\frac{2 \sqrt{\frac{\lambda^2}{4}+	an ^2 	heta}}{1-	an ^2 	heta}ight| \ \Rightarrow & \frac{\lambda^2}{4}+	an ^2 	heta=	an ^2 	heta \ \Rightarrow & \lambda=0 \end{array}$

If the angle between the lines represented by the equation $x^2+\lambda x y-y^2 \tan ^2 \theta=0$ is $2…

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