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Step-by-step worked examples β Coordinate geometry
Step-by-step solutions to past-paper-style questions on coordinate geometry , written exactly the way a tutor would explain them at the board.
1Perpendicular bisector (7 marks)
Extendedβ’ Adapted from 9709/12 May/Jun 2024β’ perpendicular bisector
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Question
Find the equation of the perpendicular bisector of the line segment joining A(1,3) and B(5,β1). (7 marks)
Step-by-step solution
Step 1
Find midpoint of AB.
M=(21+5β,23+(β1)β)=(3,1)
Step 2
Gradient of AB.
mABβ=5β1β1β3β=4β4β=β1
Step 3
Perpendicular gradient. Negative reciprocal.
mβ₯β=1
Step 4
Equation through M with gradient mβ₯β.
yβ1=1(xβ3)
Step 5
Simplify.
y=xβ2
Answer
y=xβ2.
2Equation of a circle (6 marks)
Extendedβ’ circle
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Question
Find the equation of the circle with centre (2,β1) that passes through the point (5,3). (6 marks)
Step-by-step solution
Step 1
Standard form.(xβa)2+(yβb)2=r2 where (a,b) is centre, r is radius.
Step 2
Find radius. Distance from (2,β1) to (5,3).
r=(5β2)2+(3β(β1))2β=9+16β=25β=5
Step 3
Plug in.
(xβ2)2+(y+1)2=25
Answer
(xβ2)2+(y+1)2=25.
3Line meets circle (7 marks)
Extendedβ’ intersection
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Question
Find the points where the line y=x+1 meets the circle x2+y2=13. (7 marks)
Step-by-step solution
Step 1
Substitute line into circle.
x2+(x+1)2=13
Step 2
Expand.
x2+x2+2x+1=13β2x2+2xβ12=0
Step 3
Simplify and factorise.
x2+xβ6=0β(x+3)(xβ2)=0
Step 4
Solutions.x=β3 or x=2.
Step 5
Find y values.y=x+1, so (β3,β2) and (2,3).
Answer
(β3,β2) and (2,3).
Key Formulae β Coordinate geometry
The formulae you need to memorise for coordinate geometry on the Cambridge International A Level 9709 paper, with every variable defined in plain English and a note on when to use it.
Distance formula
d=(x2ββx1β)2+(y2ββy1β)2β
(x1β,y1β),(x2β,y2β)
two points
When to use
Distance between two points; radius of circle.
Midpoint formula
M=(2x1β+x2ββ,2y1β+y2ββ)
M
midpoint of segment
When to use
Finding midpoint; centre of perpendicular bisector.
Gradient
m=x2ββx1βy2ββy1ββ
m
gradient (slope)
When to use
Slope of line through two points.
Line equation (point-slope)
yβy1β=m(xβx1β)
(x1β,y1β)
point on line
m
gradient
When to use
Equation of a line given a point and slope.
Perpendicular gradients
m1βm2β=β1
m1β,m2β
gradients of perpendicular lines
When to use
Two lines are perpendicular iff product of gradients = -1.
Circle equation (centre-radius)
(xβa)2+(yβb)2=r2
(a,b)
centre
r
radius
When to use
Standard form. From general x2+y2+Dx+Ey+F=0, complete the square.
Key Definitions and Keywords β Coordinate geometry
Definitions to memorise and the exact keywords mark schemes credit for coordinate geometry answers β sharpened from recent examiner reports for the 2026 Cambridge International A Level 9709 sitting.
Gradient
Examiner keyword
Rate of change of y with respect to x. Slope of a line.
Perpendicular
Examiner keyword
Two lines at 90Β° to each other. Gradients multiply to β1.
Perpendicular bisector
Examiner keyword
Line perpendicular to segment AB passing through midpoint of AB. Points on it are equidistant from A and B.
Circle (in coordinate geometry)
Examiner keyword
Locus of points equidistant from a fixed centre. Standard form: (xβa)2+(yβb)2=r2.
Tangent to a circle
Line touching circle at exactly one point. Perpendicular to radius at point of contact.
Common Mistakes and Misconceptions β Coordinate geometry
The traps other students keep falling into on coordinate geometry questions β taken from recent Cambridge International A Level 9709 examiner reports and mark schemes β and how to avoid them.
βUsing m1β instead of βm1β for perpendicular gradient
9709 Examiner Reports 2022-2024
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Why it happens
Forgetting the negative.
How to avoid it
Perpendicular gradient is NEGATIVE RECIPROCAL. Check: product of gradients should be β1.
βWrong sign of centre in circle equation
9709 Examiner Reports 2022-2024
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Why it happens
Confusion with βa vs a.
How to avoid it
(xβa)2+(yβb)2=r2 β centre is (a,b), NOT (βa,βb). Match signs carefully.
βForgetting square in distance formula
9709 Examiner Reports 2022-2024
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Why it happens
Rushing.
How to avoid it
d=(x2ββx1β)2+(y2ββy1β)2β β BOTH differences are squared, then ADDED, then square-rooted.
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