A ray of light is sent along the line which passes through the point (2, 3). The ray is reflected from the point P on x-axis. If the reflected ray passes through the point (6, 4), then the coordinates of P are

Let $P\equiv \left(\alpha ,0\right)$. 

Let the reflected ray makes an angle $\theta$ with +ve direction of x-axis, then the incident ray makes angle $\left(\pi -\theta \right)$ with positive direction of x-axis. 

The slope of the incident ray is

$=\frac{0-3}{\alpha -2}=\tan \left(\pi -\theta \right)$ i.e., $\tan \theta =\frac{3}{\alpha -2}$

The slope of the reflected ray is

$=\frac{4-0}{6-\alpha }=\tan \theta$ i.e., $\tan \theta =\frac{4}{6-\alpha }$

From Eq. (1) and (2), we get

$\frac{3}{\alpha -2}=\frac{4}{6-\alpha }\Rightarrow 18-3\alpha =4\alpha -8$

$\Rightarrow 7\alpha =26 or \alpha =\frac{26}{7}$

$\therefore$The coordiantes of A are $\left(\frac{26}{7},0\right)$