Coordinate Geometry of the Circle: Fluency and Exam Drill - Worksheets, Questions and Revision

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GCSE · Geometry

G2D Coordinate Geometry of the Circle: Fluency and Exam Drill

AQA 8365 · Calculator allowed · about 50 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

Write down the radius of the circle with equation x² + y² = 64.

(Total for Question 1 is 1 mark)

Write down the centre and radius of the circle (x - 3)² + (y + 2)² = 25.

(Total for Question 2 is 1 mark)

A circle has centre (0, 0) and radius 7. Write down its equation.

(Total for Question 3 is 1 mark)

Determine whether the point (5, 12) lies on the circle x² + y² = 169.

(Total for Question 4 is 2 marks)

Determine whether the point (4, 7) lies on the circle x² + y² = 50.

(Total for Question 5 is 2 marks)

Find the radius of the circle with equation x² + y² = 45, giving your answer in simplified surd form.

(Total for Question 6 is 2 marks)

Write down the centre of the circle (x + 4)² + (y - 6)² = 100.

(Total for Question 7 is 1 mark)

Find the equation of the circle with centre (2, -5) and radius 6.

(Total for Question 8 is 2 marks)

A circle has centre (0, 0) and passes through the point (6, 8). Find the equation of the circle.

(Total for Question 9 is 2 marks)

Write down the radius of the circle x² + y² = 1.

(Total for Question 10 is 1 mark)

The point (a, 7), where a > 0, lies on the circle x² + y² = 50. Find the value of a.

(Total for Question 11 is 2 marks)

Find the diameter of the circle with equation x² + y² = 169.

(Total for Question 12 is 2 marks)

A circle has equation x² + y² = 40. Determine whether the point (3, 7) lies inside, on, or outside the circle, showing your calculation.

(Total for Question 13 is 3 marks)

A circle has centre (3, 4) and passes through the point (7, 1). Find the equation of the circle.

(Total for Question 14 is 3 marks)

Find the points where the circle x² + y² = 121 crosses the x-axis.

(Total for Question 15 is 3 marks)

The line y = x + 3 intersects the circle x² + y² = 29 at two points. Find the coordinates of these points.

(Total for Question 16 is 4 marks)

A circle has equation x² + y² = 289. Find the equation of the tangent to the circle at the point (8, 15), giving your answer in the form ax + by = c.

(Total for Question 17 is 4 marks)

A circular pond has centre O at the origin of a garden's coordinate grid, where 1 unit represents 1 metre, and equation x² + y² = 49. A straight path is to be built along the line y = 2x - 3.

(a)Show that the path crosses the edge of the pond by substituting the line into the circle's equation and showing that the resulting equation has two real solutions.(2)

(b)Find the x-coordinates of the two points where the path crosses the edge of the pond, giving your answers correct to 2 decimal places.(2)

(Total for Question 18 is 4 marks)

Mark scheme · G2D Coordinate Geometry of the Circle: Fluency and Exam Drill

Question 1

B1 8 cao
Answer: radius = 8

Question 2

B1 centre (3, -2), radius 5 (both required) cao
Answer: centre (3, -2), radius 5

Question 3

B1 x² + y² = 49 cao
Answer: x² + y² = 49

Question 4

M1 substitutes to find 5² + 12² = 169
A1 concludes the point lies on the circle, since 169 = 169
Answer: Yes, the point lies on the circle

Question 5

M1 substitutes to find 4² + 7² = 65
A1 concludes the point does not lie on the circle, since 65 is not equal to 50
Answer: No, the point does not lie on the circle

Question 6

M1 radius = √45
A1 3*√5 oe cao
Answer: 3*√5

Question 7

B1 (-4, 6) cao
Answer: (-4, 6)

Question 8

M1 uses the form (x - a)² + (y - b)² = r² with a=2, b=-5, r=6
A1 (x - 2)² + (y + 5)² = 36 cao
Answer: (x - 2)² + (y + 5)² = 36

Question 9

M1 finds the radius using the distance from the origin: √6² + 8² = 10
A1 x² + y² = 100 cao
Answer: x² + y² = 100

Question 10

B1 1 cao
Answer: radius = 1

Question 11

M1 substitutes to obtain a² + 49 = 50
A1 a = 1 (taking the positive root) cao
Answer: a = 1

Question 12

M1 radius = √169 = 13
A1 diameter = 26 cao
Answer: 26

Question 13

M1 substitutes the point into x² + y² to obtain 3² + 7² = 58
M1 compares 58 with the value 40 on the right-hand side of the circle's equation
A1 concludes the point lies outside the circle, since 58 > 40
Answer: The point lies outside the circle

Question 14

M1 finds the radius using the distance formula: √(7-3)² + (1-4)²
M1 radius = √16+9 = 5, dependent on previous M1 (dM1)
A1 (x - 3)² + (y - 4)² = 25 cao
Answer: (x - 3)² + (y - 4)² = 25

Question 15

M1 substitutes y = 0 into the circle equation
M1 solves x² = 121 to find x = 11 and x = -11, dependent on previous M1 (dM1)
A1 (11, 0) and (-11, 0) (both required) cao
Answer: (11, 0) and (-11, 0)

Question 16

M1 substitutes y = x + 3 into x² + y² = 29
M1 expands and simplifies to x² + 3x - 10 = 0, dependent on previous M1 (dM1)
A1 factorises to (x + 5)(x - 2) = 0, giving x = -5 and x = 2 (both values)
A1 (-5, -2) and (2, 5) (both points, ft from the x-values)
Answer: (-5, -2) and (2, 5)

Question 17

M1 finds the gradient of the radius to (8, 15) as 15/8
M1 finds the perpendicular (tangent) gradient as -8/15
M1 forms the tangent equation using y - 15 = -(8/15)(x - 8), dependent on both previous marks (dM1)
A1 8x + 15y = 289 oe cao
Answer: 8x + 15y = 289

Question 18

(a) M1 substitutes y = 2x - 3 into x² + y² = 49 and simplifies to 5x² - 12x - 40 = 0
(a) A1 evaluates the discriminant (-12)² - 4(5)(-40) = 944, which is greater than 0, so there are two real solutions and the path crosses the pond's edge
(a) Answer: 5x² - 12x - 40 = 0, discriminant 944 > 0, so the path crosses the pond's edge at two points
(b) M1 uses the quadratic formula on 5x² - 12x - 40 = 0, ft from part (a)
(b) A1 x = 4.27 and x = -1.87 (awrt 2 decimal places, both values)
(b) Answer: x = 4.27 or x = -1.87 (awrt 2 dp)

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Coordinate Geometry of the Circle: Fluency and Exam Drill - Worksheets, Questions and Revision

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G2D Coordinate Geometry of the Circle: Fluency and Exam Drill

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