1.1: Perpendicular lines and exact area

An equation of the straight line $\displaystyle l$ is

The point $\displaystyle A$ on $\displaystyle l$ lies on the $\displaystyle y$-axis. The point $\displaystyle B$ on $\displaystyle l$ has coordinates $\displaystyle (10,b)$, where $\displaystyle b$ is an integer. The point $\displaystyle C$ divides $\displaystyle AB$ in the ratio $\displaystyle 2:3$.

The straight line $\displaystyle k$ passes through $\displaystyle C$ and is perpendicular to $\displaystyle l$.

(a) Show that an equation of $\displaystyle k$ is

The point $\displaystyle D$ with coordinates $\displaystyle (p,q)$, where $\displaystyle q$ is positive, is such that $\displaystyle AD$ is parallel to $\displaystyle k$ and the length of $\displaystyle AD$ is $\displaystyle 12\sqrt{10}$.

(b) Find the coordinates of $\displaystyle D$.

The point $\displaystyle E$ lies on $\displaystyle k$ such that $\displaystyle DE$ is parallel to the $\displaystyle y$-axis. The point $\displaystyle F$ lies on $\displaystyle l$ such that $\displaystyle DF$ is parallel to the $\displaystyle y$-axis.

(c) Find the exact area of triangle $\displaystyle ECF$.