1.9: The points and have coordinates and respectively.

The points $\displaystyle A$ and $\displaystyle B$ have coordinates $\displaystyle (-6, 8)$ and $\displaystyle (12, 2)$ respectively.

(a) Find an equation of the straight line passing through $\displaystyle A$ and $\displaystyle B$ in the form $\displaystyle ax + by + c = 0$, where $\displaystyle a, b$ and $\displaystyle c$ are integers to be found.

(b) Find the exact length of $\displaystyle AB$

The point $\displaystyle X$ with coordinates $\displaystyle (m, n)$ lies on $\displaystyle AB$ such that $\displaystyle AX:XB = 1:2$

(c) Find the value of $\displaystyle m$ and the value of $\displaystyle n$

The line $\displaystyle L$ passes through the point $\displaystyle X$ and is perpendicular to $\displaystyle AB$

The point $\displaystyle C$ with coordinates $\displaystyle (p, q)$ lies on $\displaystyle L$ where $\displaystyle p > 0$ and $\displaystyle q > 0$

Given that $\displaystyle AB$ is a diameter of a circle and $\displaystyle C$ also lies on the circumference of the circle,

(d) find

(i) the exact value of $\displaystyle p$

(ii) the exact value of $\displaystyle q$

(e) Find the exact area of triangle $\displaystyle ABC$