Find the gradient of the line joining A(2, 3) and B(6, 11).
(Total for Question 1 is 1 mark)
2
Find the midpoint of the points (4, -2) and (10, 6).
(Total for Question 2 is 1 mark)
3
Find the distance between the points (0, 0) and (6, 8).
(Total for Question 3 is 1 mark)
4
A line has gradient 3. State the gradient of a line perpendicular to it.
(Total for Question 4 is 1 mark)
5
Write down the gradient of the line with equation y = 4x - 7.
(Total for Question 5 is 1 mark)
6
Write down the y-intercept of the line with equation y = -2x + 9.
(Total for Question 6 is 1 mark)
7
A circle has equation x2 + y2 = 36. Write down the radius of the circle.
(Total for Question 7 is 1 mark)
8
Find the midpoint of the points (-5, 3) and (1, -9).
(Total for Question 8 is 1 mark)
9
Find the gradient of the line joining P(-1, 4) and Q(3, -8).
(Total for Question 9 is 1 mark)
10
A circle has centre (0, 0) and passes through the point (5, 12). Write down the radius of the circle.
(Total for Question 10 is 1 mark)
11
A circle has equation (x - 3)2 + (y + 2)2 = 49. Write down the coordinates of the centre and the radius of the circle.
(Total for Question 11 is 2 marks)
12
The points A(1, 5) and B(9, -1) are the endpoints of a diameter of a circle. Find the centre of the circle.
(Total for Question 12 is 2 marks)
13
Find an equation of the line through the point (2, -3) with gradient 4, giving your answer in the form y = mx + c.
(Total for Question 13 is 2 marks)
14
The points C(-3, 4) and D(5, -2) are joined by a straight line. Find an equation of the line CD, giving your answer in the form ax + by + c = 0 where a, b and c are integers.
(Total for Question 14 is 3 marks)
15
A circle has centre (-1, 3) and passes through the point (4, 15). Find an equation of the circle.
(Total for Question 15 is 3 marks)
16
Determine, showing your working, whether the point (8, -5) lies inside, on, or outside the circle with equation (x - 2)2 + (y + 1)2 = 50.
(Total for Question 16 is 3 marks)
17
The points E(2, 1) and F(10, 7) form a diameter of a circle. Find an equation of the circle.
(Total for Question 17 is 3 marks)
18
A circle has equation x2 + y2 - 8x + 6y - 15 = 0. Find the centre and the exact radius of the circle.
(Total for Question 18 is 4 marks)
19
A circle C has equation x2 + y2 = 45. The point A has coordinates (7, 4), which lies outside C. Find the exact length of the tangent from A to the circle.
(Total for Question 19 is 4 marks)
20
A circle has equation x2 + y2 - 4x + 6y - 19 = 0. The line l has equation y = x + k, where k is a constant. Given that l is a tangent to the circle, find the two possible values of k.
(Total for Question 20 is 4 marks)
Mark scheme · P3D Pure: Coordinate Geometry (Lines and Circles): Fluency and Exam Drill
Question 1
B1 gradient = 2 cao
Answer: 2
Question 2
B1 (7, 2) cao
Answer: (7, 2)
Question 3
B1 10 cao
Answer: 10
Question 4
B1 -1/3 cao
Answer: -1/3
Question 5
B1 4 cao
Answer: 4
Question 6
B1 (0, 9) oe cao
Answer: (0, 9)
Question 7
B1 6 cao
Answer: 6
Question 8
B1 (-2, -3) cao
Answer: (-2, -3)
Question 9
B1 -3 cao
Answer: -3
Question 10
B1 13 cao
Answer: 13
Question 11
B1 centre (3, -2) cao
B1 radius = 7 cao
Answer: centre (3, -2), radius 7
Question 12
M1 uses the midpoint formula on A and B
A1 centre = (5, 2) cao
Answer: centre = (5, 2)
Question 13
M1 uses y - (-3) = 4(x - 2) or substitutes (2,-3) into y = 4x + c
A1 y = 4x - 11 cao
Answer: y = 4x - 11
Question 14
M1 finds gradient CD = (-2-4)/(5-(-3)) = -3/4
M1 forms y - 4 = -3/4(x + 3) using point C (or point D)
A1 3x + 4y - 7 = 0 oe with integer coefficients, cao
Answer: 3x + 4y - 7 = 0
Question 15
M1 uses r2 = (4-(-1))2 + (15-3)2
A1 r2 = 169 (r = 13)
A1 (x+1)2 + (y-3)2 = 169 cao
Answer: (x + 1)2 + (y - 3)2 = 169
Question 16
M1 substitutes (8,-5) into (x-2)2 + (y+1)2, i.e. (8-2)2 + (-5+1)2
M1 compares the result to 50
A1 52 > 50 so the point lies outside the circle, cao
Answer: Outside the circle (52 > 50)
Question 17
M1 finds the centre as the midpoint of EF, (6, 4)
M1 finds the length EF and halves it to obtain the radius
A1 (x-6)2 + (y-4)2 = 25 cao
Answer: (x - 6)2 + (y - 4)2 = 25
Question 18
M1 completes the square on the x terms: x2 - 8x = (x-4)2 - 16
M1 completes the square on the y terms: y2 + 6y = (y+3)2 - 9
A1 centre = (4, -3)
A1 radius = √40 = 2sqrt(10), exact
Answer: Centre (4, -3), radius 2sqrt(10)
Question 19
B1 radius of C: r2 = 45
M1 finds OA2 = 72 + 42 = 65, where O is the centre of C
M1 uses tangent length = √OA2 - r2
A1 tangent length = √20 = 2sqrt(5) exact
Answer: Tangent length = 2sqrt(5)
Question 20
M1 substitutes y = x + k into the circle equation
A1 forms the correct quadratic in x: 2x2 + (2k+2)x + (k+3)2 - 28 = 0
M1 applies the tangency condition b2 - 4ac = 0 and simplifies to k2 + 10k - 39 = 0