Perpendicular Lines and all $x$-axis points

Question

Find the coordinates of all the points $P$ on the $x$-axis so that the line $A(1, 2)$ and $P$ is perpendicular to $B(8, 3)$ and $P$.

The answer is $(7, 0)$ and $(2, 0)$, which I understand are the x-intercepts of the two lines. How is this worked? Thanks

Answer 1

Say the coordinates of $P$ are $(x,0)$.

$AP\perp BP$ means the dot product $(1-x,2)\cdot(8-x,3)=0$.

This simplifies to $(x-1)(x-8)+6=x^2-9x+14=0$.

Can you take it from here?

Answer 2

Alternatively, you might know that the product of the slopes of two perpendicular lines is -1. The slope of the line between $(1,2)$ and $(x, 0)$ is $\frac{0-2}{x-1}$ and the slope of the line between $(8,3)$ and $(x,0)$ is $\frac{0-3}{x-8}$, so using the aforementioned perpendicularity property we have $$ \frac{0-2}{x-1} \cdot \frac{0-3}{x-8} = -1 \implies (x-1)\cdot(x-8) = -6 \implies x^2 -9x +14 = 0 $$ the same equation that J.W. Tanner got through a slightly different method.