JEE Coordinate (7)!

Any point on the line making an angle $\theta$ with the x-axis and passing through the point $(2, 3)$ can be represented by $(2+r\cos(\theta), 3+r\sin(\theta))$ where $|r|$ is the distance of the point from $(2, 3)$ .

Since we are interested in the intersection of this line with the given pair of straight lines, the above point satsfies the equation of the pair of straight lines:

$(2+r\cos(\theta))^2 - 2(2+r\cos(\theta))(3+r\sin(\theta)) - (3+r\sin(\theta))^2 = 0$

$r^2 (\sin\theta - \cos\theta)^2 + 2r(\sin\theta - \cos\theta) - 17 = 0$

Since the moduli of the roots of the above equation are the distances of the intersection points from $(2, 3)$ :

$\Bigg|\dfrac{-17}{(\sin\theta - \cos\theta)^2}\Bigg| = 17$

$\implies(\sin\theta - \cos\theta)^2=1$

$\implies 1 - 2\sin\theta\cos\theta = 1$

$\implies sin\theta\cos\theta = 0$

$\theta = \boxed{0, \dfrac{\pi}{2}}$