If the chord through (1,-2) cuts the curve 3 x^2-y^2-2 x+4 y=0 at P and Q , then the angle subtended by P Q at the origin is

Step-by-step solution

Let P Q subtend the angle at the origin. POQ = Let the slope of line is m . The equation of line is y+2=m(x-1) mx - y =2+ m mx - y 2+ m =1 ...(i) The given curve is 3 x^2-y^2-2 x+4 y=0 ...(ii) Homogenising eq ^ n (ii), we get 3 x^2-y^2-2 x(1)+4 y(1)=0 aligned & 3 x^2-y^2-2 x ( m x-y 2+m )+4 y ( m x-y 2+m )=0 & 3(2+m) x^2-(2+m)-2 m x^2+2 x y+4 m x y-4 y^2=0 & x^2(6+3 m-2 m)-y^2(2+m+4)+y(4 m+2)=0 & (6+m) x^2-(6+m) y^2+(4 m+2) x y=0 aligned Now, Coefficient of x^2+ Coefficient of y^2=(6+m)-(6+m)=0 The angle subtended