CAIE A-Level Mathematics 1.3 Coordinate geometry

[Maximum number: 2]

Figure for Question 1.3 Coordinate geometry question 1 — CAIE A-Level Mathematics AS

The diagram shows the curve with equation $x=y^{2}+1$ . The points A(5,2) and B(2,-1) lie on the curve.

Find an equation of the line A B.

Gradient of $A B=\frac{2-(-1)}{5-2}$

M1

Expect 1, must be from $\Delta y / \Delta x$ .

Equation of A B is y-2=1(x-5) or y+1=1(x-2)

A1

OE. Expect y=x-3.

[Maximum number: 10]

Points A(-2,3), B(3,0) and C(6,5) lie on the circumference of a circle with centre D.

(a)

Show that angle $A B C=90^{\circ}$ .

[ 2 ]

Gradient of $A B=-\frac{3}{5}$ , gradient of $B C=\frac{5}{3}$ or lengths of all 3 sides or vectors
M1
Attempting to find required gradients, sides or
vectors
$m_{a b} m_{b c}=-1$ or Pythagoras or $\overrightarrow{A B} \cdot \overrightarrow{B C}=0$ or $\cos A B C=0$ from cosine rule
A1
WWW

(b)

Hence state the coordinates of D.

[ 1 ]

(c)

Find an equation of the circle.

[ 2 ]

$\left(x-\text { their } \mathrm{x}_{\mathrm{c}}\right)^{2}+\left(y-\text { their } y_{c}\right)^{2}\left[=r^{2}\right]$ or $\left(\text { their } x_{\mathrm{c}}-x\right)^{2}+\left(\text { their } y_{c}-y\right)^{2}=\left[r^{2}\right]$
M1
Use of circle equation with their centre
$(x-2)^{2}+(y-4)^{2}=17$
A1
Accept $x^{2}-4 x+y^{2}-8 y+3=0$ OE

(d)

The point E lies on the circumference of the circle such that B E is a diameter.

Find an equation of the tangent to the circle at E.

[ 5 ]

\left(\frac{x+3}{2}, \frac{y+0}{2}\right)=(2,4) \text { or } \mathbf{B E}=2 \mathbf{B D}=2\binom{-1}{4}
Or Equation of B E is y=-4(x-3) or y-4=-4(x-2) leading to y=-4 x+12 Substitute equation of B E into circle and form a 3-term quadratic.
M1
Use of mid-point formula, vectors, steps on a diagram
May be seen to find x coordinate at E
(x, y)=(1,8) or $\mathbf{O E}=\binom{3}{0}+\binom{-2}{8}=\binom{1}{8}$
A1
E=(1,8)
Accept without working for both marks SC B2
Gradient of B D, m,=-4 or gradient $A C=\frac{1}{4}=$ gradient of tangent
B1
Or gradient of B E=-4
Equation of tangent is y-8=1 / 4(x-1) OE
M1 A1
For M1, equation through their E or ( 1,8 ) (not,
A, B or C ) and with gradient $\frac{-1}{\text { their }-4}$

CAIE A-Level Mathematics 1.3 Coordinate geometry Question Bank

Exam points

Covered syllabus points

Covered syllabus points

1.3 Coordinate geometry question 1

1.3 Coordinate geometry question 2

Question (a)

Question (b)

Question (c)

Question (d)