Find the coordinates of all points P on the x-axis so that the line through A (1, 2) and P is perpendicular to the line through B (8, 3) and P
Let P have coordinates (a, 0).
Get the gradient $\displaystyle m_{AP}$ of the line passing through A and P in terms of a.
Get the gradient $\displaystyle m_{BP}$ of the line passing through B and P in terms of a.
Solve the equation $\displaystyle m_{AP} \cdot m_{BP} = -1$ for a.