Grade 12 Maths · Analytical Geometry

Use the equation of a circle

Use the circle equation (x − a)² + (y − b)² = r², for example centre (2; −1), radius 3 gives (x − 2)² + (y + 1)² = 9.

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Equation of Circle Grade 12Kevinmathscience

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Analytical Geometry Grade 12: L3 Convert equation of circle to centre- radius formOnline Maths by Miss Pythagoras

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Question 4.6 · May/June 2025 · Paper 2 · National (DBE) · 3 marks

In the diagram, M is the centre of the circle having equation

(x - 3)^{2} + y^{2} = 25

E (- 1; 3)

and T are points on the circle.

EC is a tangent to the circle at E and cuts the x-axis at D.

E D = \frac{15}{4}

units. MT is produced to meet the tangent at

C (- 7; p)

4.6 If the radius of the circle, centred at M, is increased by

7

units, determine whether S lies inside or outside the new circle. Support your answer with the necessary calculations.(3)

Work it out on paper first. Then mark yourself with the official memo, one tick at a time.

More past paper questions: use the equation of a circle

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3. Question WE11 · Siyavula Grade 12 Mathematics, 7.2 Equation of a circle, Worked example 11 · 5 marks
WE11 A circle with centre on the line $y = - x + 5$ passes through the points $P (5; 8)$ and $Q (9; 4)$ . Determine the equation of the circle.(5)
Show the memo (5 ticks)Memo
1. ✓
  General equation with centre on the line $(x - a)^{2} + (y - b)^{2} = r^{2}$
  The centre lies on $y = - x + 5$ , so it is $(a; - a + 5)$ :
  $(x - a)^{2} + (y + a - 5)^{2} = r^{2}$
  1 mark · Use the equation of a circle
2. ✓
  Substitute P $(5 - a)^{2} + (8 + a - 5)^{2} = r^{2}$
  $25 - 10 a + a^{2} + a^{2} + 6 a + 9 = r^{2}$
  $2 a^{2} - 4 a + 34 = r^{2} \dots (1)$
  1 mark · Use the equation of a circle
3. ✓
  Substitute Q $(9 - a)^{2} + (4 + a - 5)^{2} = r^{2}$
  $81 - 18 a + a^{2} + a^{2} - 2 a + 1 = r^{2}$
  $2 a^{2} - 20 a + 82 = r^{2} \dots (2)$
  1 mark · Use the equation of a circle
4. ✓
  Solve for a $(1) - (2)$ : $16 a - 48 = 0$
  $∴ a = 3$
  1 mark · Solve simultaneous equations by elimination
5. ✓
  Equation $r^{2} = 2 (3)^{2} - 20 (3) + 82 = 40$
  $b = - a + 5 = 2$
  The equation of the circle is $(x - 3)^{2} + (y - 2)^{2} = 40$ .
  1 mark · Use the equation of a circle
4. Question WE8 · Siyavula Grade 12 Mathematics, 7.2 Equation of a circle, Worked example 8 · 4 marks
WE8 Determine the coordinates of the centre of the circle and the length of the radius for $3 x^{2} + 6 x + 3 y^{2} - 12 y - 33 = 0$ .(4)
Show the memo (4 ticks)Memo
1. ✓
  Coefficients equal to 1Take out $3$ as a common factor:
  $x^{2} + 2 x + y^{2} - 4 y - 11 = 0$
  1 mark · Complete the square
2. ✓
  Complete the square $(\frac{2}{2})^{2} = 1$ and $(\frac{- 4}{2})^{2} = 4$
  $(x^{2} + 2 x + 1) - 1 + (y^{2} - 4 y + 4) - 4 - 11 = 0$
  $(x + 1)^{2} + (y - 2)^{2} = 16$
  1 mark · Complete the square
3. ✓
  CentreThe centre of the circle is $(- 1; 2)$ .
  1 mark · Use the equation of a circle
4. ✓
  Radius $r^{2} = 16$ , so the radius is $4$ units.
  1 mark · Use the equation of a circle
5. Question WE6 · Siyavula Grade 12 Mathematics, 7.2 Equation of a circle, Worked example 6 · 3 marks
WE6 Given a circle with centre $O (0; 0)$ and a radius of $45$ units. Determine the possible coordinates of the point(s) on the circle which have an $x$ -value that is twice the $y$ -value.(3)
Show the memo (3 ticks)Memo
1. ✓
  Equation of the circle $x^{2} + y^{2} = r^{2}$
  $x^{2} + y^{2} = (45)^{2}$
  $x^{2} + y^{2} = 45$
  1 mark · Use the equation of a circle
2. ✓
  Substitute x = 2y $(2 y)^{2} + y^{2} = 45$
  $5 y^{2} = 45$
  $y^{2} = 9$
  $∴ y = \pm 3$
  1 mark · Solve linear and non-linear simultaneous equations
3. ✓
  PointsIf $y = - 3$ : $x = 2 (- 3) = - 6$
  If $y = 3$ : $x = 2 (3) = 6$
  The points are $(6; 3)$ and $(- 6; - 3)$ .
  1 mark · Solve linear and non-linear simultaneous equations

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