5.7 SPM Practice (Long Questions 5)

Question 9:


The diagram above shows that the straight line PMQ intersects with PNR at N. Given that OQ = OR and M is the midpoint of PQ. Find
(a) the coordinate of P
(b) the value of m and n.

Solution:
$\begin{array}{l}\text{(a)}\\ \text{Given }M\text{ is midpoint of }PQ\\ x\text{-coordinate of }P=0\\ \text{For }y\text{-coordinate of }P,\\ \frac{y+0}{2}=3\\ y=6\\ \text{Coordinates of }P=\left(0,6\right).\end{array}$

$\begin{array}{l}\text{(b)}\\ \frac{0+4}{2}=m\\ 2m=4\\ m=2\\ \\ \text{Gradient of }PR\\ =\frac{6-0}{0-\left(-4\right)}\\ =\frac{6}{4}\\ =\frac{3}{2}\\ \\ \text{At point }N\left(n,4\right)\text{, }\\ \text{using }{y}_1=m{x}_1+c\\ 4=\frac{3}{2}n+6\\ 8=3n+12\\ 3n=-4\\ n=-\frac{4}{3}\end{array}$

Question 10 (5 marks):
Diagram 6 shows two parallel straight lines, POQ and RMS drawn on a Cartesian plane.

Diagram 6

Find
(a) the equation of the straight line RMS,
(b) the x-intercept of the straight line MN.


Solution:
(a)
$\begin{array}{l}{m}_{RMS}=m_{POQ}\\ =\frac{1-0}{-2-0}\\ =-\frac{1}{2}\\ \\ \text{At point }\left(4,6\right)\\ y_1=m{x}_1+c\\ 6=-\frac{1}{2}\left(4\right)+c\\ 6=-2+c\\ c=8\\ \\ \text{The equation of straight line }RMS\text{ is}\\ y=-\frac{1}{2}x+8\end{array}$


(b)
$\begin{array}{l}\text{When }y=0,\\ -\frac{1}{2}x+8=0\\ \frac{1}{2}x=8\\ x=16\\ \\ x\text{-intercept of straight line }MN\text{ is }16.\end{array}$