Question

The angle between lines joining the origin to the points of intersection of the line $\sqrt{3}x+y=2$ and the curve $y^2-x^2=4$, is

• A.
  
• B.
  
• C.
  
• D.

Answer 1

The correct option is C


Making $y^2-x^2=4$ homogeneous with the help of the straight line $\sqrt{3}x+y=2$, combined equation of lines joining the points of intersection of the curve $y^2-x^2=4$ and the given line to the origin is given by
$y^2-x^2=4\left(\frac{\sqrt{3}x+y}{2}\right)$
$\Rightarrow$ $4x^2-2\sqrt{3}xy+0.y^2=0$
If $\theta$ is the angle between these lines, then
$\theta =ta{n}^{-1}\left|\frac{2\sqrt{h^2-ab}}{a+b}\right|=ta{n}^{-1}\left|\frac{2\sqrt{3-0}}{4+0}\right|$
Here, $a=4,h=\sqrt{3},b=0$
$\theta =ta{n}^{-1}\left(\frac{\sqrt{3}}{2}\right)$