1.13: Past-paper question 4

The point $\displaystyle A$ with coordinates $\displaystyle (12, 14)$ and the point $\displaystyle B$ with coordinates $\displaystyle (q, 2)$ where $\displaystyle q$ is a constant, lie on the straight line with equation $\displaystyle 3y - 2x - p = 0$ where $\displaystyle p$ is a constant.

(a) Find the value of $\displaystyle p$ and the value of $\displaystyle q$

The line $\displaystyle L$ is perpendicular to $\displaystyle AB$ and passes through the point $\displaystyle X$, which lies on $\displaystyle AB$ such that $\displaystyle AX : XB = 1 : 2$

(b) Find an equation for $\displaystyle L$ in the form $\displaystyle ax + by + c = 0$ where $\displaystyle a, b$ and $\displaystyle c$ are integers to be found.