The combined equation of two lines through the origin and making an angle of $45^{\circ}$ with the line $3…

The combined equation of two lines through the origin and making an angle of $45^{\circ}$ with the line $3 x+y=0$, is
  1. $2 x^2-3 x y-2 y^2=0$
  2. $2 x^2+3 x y+4 y^2=0$
  3. $2 x^2+3 x y-2 y^2=0$
  4. $2 x^2-3 x y+2 y^2=0$

Solution

Given line $3 x+y=0 \Rightarrow$ slope $=-3$ Let the slope of required line be $m$ $\begin{aligned} & \therefore \quad \tan 45^{\circ}=\left|\frac{m+3}{1-3 m}\right| \\ & \quad \Rightarrow 1=\left|\frac{m+3}{1-3 m}\right| \\ & \quad \Rightarrow 2 m^2-3 m-2=0...(i) \end{aligned}$
Since the line passes through origin, its equation is $y=m x$ $\Rightarrow \mathrm{m}=\frac{y}{x}$
Substituting the value of $m$ in equation (i), we get $\begin{aligned} & 2\left(\frac{y}{x}\right)^2-3\left(\frac{y}{x}\right)-2=0 \\ & \Rightarrow 2 y^2-3 x y-2 x^2=0 \\ & \Rightarrow 2 x^2+3 x y-2 y^2=0 \end{aligned}$

Asked in: MHT CET 2024 (04 May Shift 2)

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