A chord through the point $(1,-2)$ cuts the curve $3 x^2-y^2-2 x+4 y=0$ at $P$ and $Q$. If $P Q$ subtends an angle $ heta$ at the origin, then $ heta$ equals

Question

A chord through the point $(1,-2)$ cuts the curve $3 x^2-y^2-2 x+4 y=0$ at $P$ and $Q$. If $P Q$ subtends an angle $	heta$ at the origin, then $	heta$ equals

Aicharya · Accepted Answer

Given curve, $3 x^2-y^2-2 x+4 y=0$ Chord passes through $(1,-2)$ cut the given curve at $P$ and $Q$

Let slope of line $P Q$ is $m$. Then, equation of line is $ \begin{aligned} & y-y_1=m\left(x-x_1\right) \\ & \Rightarrow \quad y+2=m(x-1) \\ & y+2=m x-m \\ & \Rightarrow \quad \frac{m x-y}{2+m}=1 \\ & \end{aligned} $ Use Eq. (ii) in Eq. (i) as follows, $ \begin{aligned} & 3 x^2-y^2-2 x\left(\frac{m x-y}{2+m}\right)+4 y\left(\frac{m x-y}{2+m}\right)=0 \\ & \Rightarrow 3(2+m) x^2-(2+m) y^2-2 x(m x-y) \\ &+ 4 y(m x-y)=0 \\ & \Rightarrow x^2(6+3 m-2 m)-y^2(2+m+4)+x y \\ &(4 m+2)=0 \end{aligned} $ General equation of curve is $ a x^2+2 h x y+b y^2+2 g x+2 f y+c=0 $ Compare Eqs. (iii) and (iv), $ a=6+m, b=-6-m $ This gives, $a+b=0$ $\Rightarrow$ Angle subtended by these two lines at origin is $90^{\circ}$

A chord through the point $(1,-2)$ cuts the curve $3 x^2-y^2-2 x+4 y=0$ at $P$ and $Q$. If $P Q$ subtends an…

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