Short Questions (Question 8 & 9)

Question 8 (3 marks):
The following information refers to the equation of two straight lines, AB and CD.

$\boxed{\begin{array}{l}AB:y-2kx-3=0\\ CD:\frac{x}{3h}+\frac{y}{4}=1\\ \\ \text{where }h\text{ and }k\text{ are constants}\text{.}\end{array}}$

Given the straight lines AB and CD are perpendicular to each other, express h in terms of k.

Solution:
$\begin{array}{l}AB:y-2kx-3=0\\ y=2kx+3\\ m_{AB}=2k\\ \\ CD:\frac{x}{3h}+\frac{y}{4}=1\\ m_{CD}=-\frac{4}{3h}\\ \\ m_{AB}\times m_{CD}=-1\\ 2k\times \left(-\frac{4}{3h}\right)=-1\\ -8k=-3h\\ h=\frac{8}{3}k\end{array}$

Question 9 (3 marks):
A straight line passes through P(3, 1) and Q(12, 7). The point R divides the line segment PQ such that 2PQ = 3RQ.
Find the coordinates of R.

Solution:



$\begin{array}{l}2PQ=3RQ\\ \frac{PQ}{RQ}=\frac{3}{2}\\ \\ \text{Point }R\\ =\left(\frac{1\left(12\right)+2\left(3\right)}{1+2},\frac{1\left(7\right)+2\left(1\right)}{1+2}\right)\\ =\left(\frac{18}{3},\frac{9}{3}\right)\\ =\left(6,3\right)\end{array}$