Parallel lines: IGCSE Mathematics questions with answers
Free Parallel lines questions for IGCSE Mathematics (0580), part of Coordinate geometry. Each question has the answer and the marking points an examiner would look for.
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 2Structured · 4 marksA* challenge
The line L has gradient −2 and passes through the point (2, −6). 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
1
y = −2x − 2 M1 B = (0, −2), A = (−1, 0) A1 ½ × base × height M1 1 A1
4
Question 3Structured · 6 marks
The diagram shows the line L.
Fig. 3.1
(a)
Find the gradient of line L.
[2]
(b)
Find the equation of line L in the form y = mx + c.
[2]
(c)
Find the equation of the line parallel to L that passes through the point (−4, 2).
[2]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
12
48 using two points on the line M1 12A1
2
(b)
y = 12x − 2
y = 12x − 2 B2 B1 for correct gradient or correct intercept in y = mx + c
2
(c)
y = 12x + 4
2 = 12 × −4 + c M1 y = 12x + 4 A1
2
Question 4Structured · 2 marks
Show that the lines 3y = 18x + 1 and y − 6x = 7 are parallel.
[2]
Show answers and mark scheme
Part
Answer and marking points
Marks
both gradient 6
first line gradient 6 M1 second line gradient 6, so parallel A1
2
Question 5Structured · 4 marks
The equation of a line is 3x + 6y = 36.
(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)
−12
rearranges to y = … M1 −12A1
2
(b)
(12, 0) and (0, 6)
(12, 0) B1 (0, 6) B1
2
Question 6Structured · 4 marksA* challenge
The line L has gradient 3 and passes through the point (2, 15). 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
272
y = 3x + 9 M1 B = (0, 9), A = (−3, 0) A1 ½ × base × height M1 272A1
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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