1.14: Past-paper question 6

The line $\displaystyle l$ passes through the point $\displaystyle A$ with coordinates $\displaystyle (-2, 2)$ and the point $\displaystyle B$ with coordinates $\displaystyle (3, 12)$

The point $\displaystyle C$ with coordinates $\displaystyle (p, q)$ lies on $\displaystyle l$ such that $\displaystyle AC : CB = 3 : 2$

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

The line $\displaystyle k$ is perpendicular to $\displaystyle l$ and passes through the point $\displaystyle C$

(b) Show that an equation of $\displaystyle k$ is $\displaystyle 2y + x - 17 = 0$

The line $\displaystyle k$ crosses the $\displaystyle x$-axis at the point $\displaystyle D$

(c) Find the exact length of $\displaystyle CD$

The point $\displaystyle X$ with coordinates $\displaystyle (m, n)$ lies on $\displaystyle l$ such that

area of triangle $\displaystyle DXC = 80$ units2

Given that $\displaystyle m > 0$

(d) find the value of $\displaystyle m$ and the value of $\displaystyle n$