Coordinate Geometry: IGCSE Mathematics questions and answers
10 free IGCSE Mathematics Coordinate Geometry questions written in the style of Cambridge Mathematics papers, each with its answer and mark scheme. 4 of them are A* challenge questions, built to catch the mistakes strong students still make.
This topic covers Gradient & equation of a line, Length & midpoint, Parallel lines, Perpendicular lines. Work through the questions below, then generate a full Coordinate Geometry paper with a fresh set every time.
A is the point (−3, 1) and B is the point (−7, 7).
(a)
Find the equation of the perpendicular bisector of AB. Give your answer in the form ax + by = c, where a, b and c are integers.
[5]
(b)
The perpendicular bisector crosses the x-axis at P. Find the x-coordinate of P.
[1]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
2x − 3y = −22
midpoint (−5, 4) B1 gradient AB = −32B1 perpendicular gradient = 23M1 substitute midpoint into y = mx + c (or equivalent) M1 2x − 3y = −22 oe A1
5
(b)
−11
−11 B1
1
Question 2Structured · 4 marksA* challenge
The line L has gradient −3 and passes through the point (2, −11). L crosses the x-axis at A and the y-axis at B. O is the origin. Find the area of triangle OAB.
[4]
Show answers and mark scheme
Part
Answer and marking points
Marks
256
y = −3x − 5 M1 B = (0, −5), A = (−53, 0) A1 ½ × base × height M1 256A1
4
Question 3Structured · 11 marksA* challenge
The line L has equation y = 2x. A is the point (1, 7) and B is the point (3, 9). A and B are on the same side of L.
(a)
Find the coordinates of the point on L that is nearest to A, and the exact shortest distance from A to L.
[3]
(b)
P is a point on L. Find the coordinates of P for which AP + PB is as small as possible, and the exact value of this least distance.
[5]
(c)
Q is a point on L. Find the coordinates of Q for which QA − QB is as large as possible, and the exact value of this greatest difference.
[3]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
(3, 6); √5
line through A perpendicular to L: y − 7 = −½(x − 1) M1 meets y = 2x at (3, 6) A1 √(22 + 12) = √5 A1
3
(b)
P = (3.75, 7.5); least AP + PB = 2√5
reflects A in L: A′ = (5, 5) M1 A′ = 2(3, 6) − (1, 7) A1 least AP + PB is when P is on A′B: y = −2x + 15 M1 P = (3.75, 7.5) A1 A′B = √(22 + 42) = 2√5 A1
5
(c)
Q = (6, 12); 2√2
QA − QB ≤ AB, with equality when Q, B, A are collinear: line AB is y = x + 6 M1 Q = (6, 12) A1 AB = 2√2 A1
3
Question 4Structured · 9 marksA* challenge
The circle x2 + y2 = 50 and the line y = x + 6 intersect at P and Q.
(a)
Find the coordinates of P and Q.
[4]
(b)
Find the exact length of PQ, giving your answer in the form k√2 or k√n.
[2]
(c)
Find the equation of the tangent to the circle at P.
[3]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
(1, 7) and (−7, −1)
substitute the line into the circle M1 three-term quadratic = 0 A1 solve M1 both points A1
4
(b)
8√2
√(82 + 82) M1 8√2 A1
2
(c)
y = −17x + 507
radius gradient 7 M1 perpendicular gradient −17M1 equation through P A1
3
Question 5Structured · 4 marks
A is the point (−1, −3) and B is the point (−4, −5).
(a)
Find the coordinates of the midpoint of AB.
[2]
(b)
Find the length of AB.
[2]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
(−2.5, −4)
One coordinate correct B1 Both correct B1
2
(b)
3.61
√(32 + 22) M1 3.61 A1
2
Question 6Structured · 4 marks
The equation of a line is 3x + 4y = 24.
(a)
Find the gradient of the line.
[2]
(b)
Find the coordinates of the points where the line crosses the x-axis and the y-axis.
[2]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
−34
rearranges to y = … M1 −34A1
2
(b)
(8, 0) and (0, 6)
(8, 0) B1 (0, 6) B1
2
Question 7Structured · 4 marks
A is the point (−2, 3) and B is the point (0, −1).
(a)
Find the gradient of AB.
[2]
(b)
Find the equation of the line AB in the form y = mx + c.
[2]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
−2
−42M1 −2 A1
2
(b)
y = −2x − 1
substitutes a point into y = mx + c M1 y = −2x − 1 A1
2
Question 8Structured · 5 marks
The vertices of triangle ABC are A(0, −1), B(3, 3) and C(6, −1).
(a)
Show that the triangle is isosceles.
[3]
(b)
Find the area of the triangle.
[2]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
AB = BC
AB = √(32 + 42) M1 BC = √(32 + 42) M1 AB = BC so isosceles A1
3
(b)
12
½ × 6 × 4 M1 12 A1
2
Question 9Structured · 2 marks
A line L has equation y = 4x + 2.
Find the equation of the line parallel to L that passes through (−1, −5).
[2]
Show answers and mark scheme
Part
Answer and marking points
Marks
y = 4x − 1
gradient 4 M1 y = 4x − 1 A1
2
Question 10Structured · 2 marks
Show that the lines 2y = 8x + 1 and y − 4x = 7 are parallel.
[2]
Show answers and mark scheme
Part
Answer and marking points
Marks
both gradient 4
first line gradient 4 M1 second line gradient 4, so parallel A1
In the 0580 syllabus, Coordinate Geometry covers Gradient & equation of a line, Length & midpoint, Parallel lines, Perpendicular lines.
Where can I get Coordinate Geometry questions with mark schemes?
This page has 10 Coordinate Geometry questions with answers. For more, the free generator on mentorriyaz.in builds a new Coordinate Geometry paper with a full mark scheme each time you click.
Want a mentor, not just a paper?
Riyaz Shaikh has mentored IGCSE, IB and A Level students for 15+ years, one-on-one, online and in Mumbai. The hardest questions on this site are the ones he uses to find a student's weak points.
Original Cambridge-style practice questions written for mentorriyaz.in. These are not Cambridge past paper questions, and this site is not affiliated with or endorsed by Cambridge University Press & Assessment. Updated 2026-11-04. Paper generator · mentorriyaz.in