1.16: Past-paper question 9

The point $\displaystyle A$ has coordinates $\displaystyle (-4, 3)$ and the point $\displaystyle B$ has coordinates $\displaystyle (6, 8)$
The points $\displaystyle A$ and $\displaystyle B$ lie on the line $\displaystyle k$

(a) Find an equation of $\displaystyle k$

The point $\displaystyle C$, on $\displaystyle k$, is such that $\displaystyle AC : CB = 4:1$

(b) Find the coordinates of point $\displaystyle C$

The point $\displaystyle D$ with coordinates $\displaystyle (p, q)$, where $\displaystyle p < 0$, lies on the line $\displaystyle l$ through $\displaystyle C$ that is perpendicular to $\displaystyle k$

The length of $\displaystyle CD$ is $\displaystyle 8\sqrt{5}$

(c) Find the coordinates of $\displaystyle D$

(d) Find the area of triangle $\displaystyle ACD$