9.3.5 Straight Lines, PT3 Focus Practice

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:
Diagram 10 shows two parallel straight lines, JK and LM, drawn on a Cartesian plane.
The straight line KM is parallel to the x-axis.

Diagram 10

Find
(a) the equation of the straight line KM,
(b) the equation of the straight line LM,
(c) the value of k.

Solution:
(a)
The equation of the straight line KM is y = 3

(b)
$\begin{array}{l}\text{Given, equation of }JK:\\ 2y=4x+3\\ y=2x+\frac{3}{2}\\ \text{Thus, }{m}_{JK}=2\\ m_{LM}=m_{JK}=2\\ \\ y=mx+c\\ \text{At }M\left(5,3\right)\\ 3=2\left(5\right)+c\\ 3=10+c\\ c=-7\\ \\ \therefore \text{ Equation of the straight line }LM\\ \text{is }y=2x-7.\end{array}$


(c)
$\begin{array}{l}\text{Substitute }\left(k,0\right)\text{ into }y=2x-7\\ 0=2\left(k\right)-7\\ 7=2k\\ k=\frac{7}{2}\end{array}$