The combined equation of the lines passing through the origin making an acute angle $\propto$ with the line $y=x$ is

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Aicharya · Accepted Answer

The required lines are $\begin{aligned} & y=\frac{1+	an \alpha}{1-	an \alpha} x 	ext { and } y=\frac{1-	an \alpha}{1+	an \alpha} x \ & \Rightarrow(1+	an \alpha) y=(1+	an x) x 	ext { and }(1+	an \alpha) y=(1-	an \alpha) x \ & 	ext { joint equation } \ & \{1+	an \alpha) x-(1-	an \alpha) y\}\{1-	an \alpha) x-(1+	an \alpha) y\}=0 \ & \left.\Rightarrow(1-	an 2 \alpha) x^2-\left\{(1+	an \alpha)^2ight)+(1-	an \alpha)^2ight\} x y+\left(1-	an ^2 \alphaight) y^2=0 \ & \Rightarrow x^2-2\left(\frac{1+	an ^2 \alpha}{1-	an ^2 \alpha}ight)^{x y+y^2=0} \ & \Rightarrow x^2-\frac{2}{\cos 2 \alpha} x y+y^2=0 \ & \Rightarrow x^2-2 \sec 2 a x y+y^2=0\end{aligned}$

The combined equation of the lines passing through the origin making an acute angle $\propto$ with the line…

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