Start by sketching a diagram of the information given, to help you visualise the problem and break it down into smaller steps
You don't have to include the axes, but try to put the points in roughly the correct place according to the coordinates, and show a right-angle at angle ABC

oBLCtCXj_edexcel-tq-mis-2-13-vh-8

any sensible sketch [1]

To find the missing value $k$ we can find the gradient of AB and substitute it into the gradient formula for the perpendicular gradient of BC

The gradient of AB using the gradient formula, $m = \frac{y_{1} - y_{2}}{x_{1} - x_{2}}$ , is

$m_{A B} = \frac{1 - 5}{- 2 - 6} = \frac{- 4}{- 8} = \frac{1}{2}$

AB and BC are perpendicular therefore their gradients are negative reciprocals therefore

$m_{B C} = - 2$

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Now we can substitute $m = - 2$ , $(6, 5)$ and $(4, k)$ into the gradient formula for BC and solve to find $k$

$\begin{array}{rcl} - 2 & = & \frac{5 - k}{6 - 4} \\ - 2 & = & \frac{5 - k}{2} \\ - 4 & = & 5 - k \\ k & = & 9 \end{array}$

[1]

Now we know that $C$ is $(4, 9)$ and we can find the equation of a line through $A (- 2, 1)$ and $C (4, 9)$
First find the gradient

$m_{A C} = \frac{1 - 9}{- 2 - 4} = \frac{- 8}{- 6} = \frac{4}{3}$

Substitute either $A (2, - 1)$ or $C (4, 9)$ into $y = \frac{4}{3} x + c$ and solve to find $c$ . Here we will use $C (4, 9)$ (as it has no negative numbers)

$\begin{array}{rcl} 9 & = & \frac{4}{3} (4) + c \\ 9 & = & \frac{16}{3} + c \\ c & = & 9 - \frac{16}{3} \\ = & \frac{27}{3} - \frac{16}{3} \\ = & \frac{11}{3} \end{array}$

Now we can write the equation of the line through A and C

$\begin{array}{rcl} y & = & \frac{4}{3} x + \frac{11}{3} \end{array}$

[1]

Finally we can rearrange into the form $a y + b x = c$ . Start by multiplying all terms by 3 to clear the fractions, then collect the terms in $x$ and $y$ on the left hand side

$\begin{array}{rcl} 3 y & = & 4 x + 11 \\ 3 y - 4 x & = & 11 \end{array}$

$\begin{array}{rcl} 3 y - 4 x & = & 11 \end{array}$ [1]

$x$	$- 2$	$- 1$	$0$	$1$	$2$
$y$	$1$		$5$

Linear Graphs y = mx + c (OCR GCSE Maths)

Exam Questions