CAIE IGCSE Additional Math 7 straight-line graphs

Use this Straight-line Graphs hub to move from line equations and gradients into perpendicular lines, midpoints, lengths and bisectors.

Covered syllabus points

7.1. Use straight-line equations 7.2. Use parallel and perpendicular gradients 7.3. Solve midpoint, length and bisector problems 7.4. Transform relationships to straight-line form

[Maximum number: 7]

(a)

A straight line passes through the points (4,23) and (-8,29). Find the point of intersection, P, of this line with the line y=2x+5.

[ 5 ]

$y=-\frac12x+25\text{ isw}$
B3.

M1 for $m=\frac{29-23}{-8-4}$ oe or $-\frac12$ , and M1 FT for $\frac{y-23}{x-4}=$ their $-\frac12$ oe, or y= their $-\frac12x+c$ and $23=-\frac12\times4+c$ oe.

OR M1 for solving 23=4m+c, 29=-8m+c for $m=-\frac12$ or c=25, and M1 FT for correctly using their m or their c to find c or m.

Solves their linear equation simultaneously with y=2x+5 to find x or y. M1: FT their $y=-\frac12x+25$ oe.

(8,21). A1.

(b)

Find the distance of P from the origin.

[ 2 ]

$\sqrt{8^2+21^2}\text{ oe}$
M1: FT their (8,21).

$\sqrt{505}$ isw, or 22.5, or $22.47[22\ldots]$ rot to 2 or more dp. A1.

[Maximum number: 10]

The coordinates of points A and B are (-5,6) and (4,-6) respectively. The point C lies on the line AB, between A and B, such that $\frac{AC}{CB}=\frac12$ .

(a)

The line CD is perpendicular to AB. Find the equation of CD in the form y=mx+c.

[ 4 ]

$m_{AB}=\frac{-6-6}{4-(-5)}=-\frac43$
B1
$m_{CD}=\frac34$
M1
$y-2=\frac34(x+2)$
M1
$y=\frac34x+\frac72$
A1

(b)

The length of BD is $\sqrt{125}$ . Find the coordinates of the two possible positions of point D.

[ 6 ]

$(x-4)^2+(y+6)^2=125$
B1
Use $y=\frac34x+\frac72$ to eliminate one unknown
M1
$(x-4)^2+\left(\frac34x+\frac72+6\right)^2=125$
A1
$25x^2+100x-300=0$
A1
Factorises or solves the quadratic
A1
$(2,5)\quad\text{and}\quad(-6,-1)$
A1

[Maximum number: 2]

Point A has coordinates (3,-1).
A circle has equation $(x-4)^{2}+(y+3)^{2}=5$ .

Given that AB is a diameter of the circle, find the coordinates of B.

$\binom{4}{-3}+\binom{1}{-2} \text { oe, soi }$

or using centre and point A e.g.
$\frac{3+x}{2}=4 \text { and } \frac{-1+y}{2}=-3$

or solve simultaneously the line y=-2 x+5 with the circle leading to 3 term quadratic $x^{2}-8 x+15[=0]$ with an attempt to solve

M1

Award M1 for the 2 equations

Award the M1 for the quadratic

(5,-5)

A1

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Covered syllabus points

7. Straight-line graphs question 1

Question (a)

Question (b)

7. Straight-line graphs question 2

Question (a)

Question (b)

7. Straight-line graphs question 3