Grade 12 Maths · Analytical Geometry · Paper 2

Equation of a Circle Grade 12

Real Equation of a Circle past paper questions and answers from 2022 to 2025, with each memo split into the ticks examiners give. Try one, mark yourself, and see exactly which skill cost you marks.

past paper questions: 53 past paper questions
skills: 13 skills
videos: 13 videos

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Question 1.2 · May/June 2025 · Paper 1 · National (DBE) · 6 marks

1.2 A rectangle having sides of

(y - 3)

metres and

(x + 2)

metres has a perimeter of

24

metres and an area of

32

square metres. Calculate the values of

x

and

y

.(6)

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

Equation of a Circle skills

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More Equation of a Circle Grade 12 questions and answers

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4. Question 4.2 · May/June 2025 · Paper 2 · National (DBE) · 4 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 $C (- 7; p)$ .
4.2 Determine the equation of the tangent EC in the form $y = m x + c$ .(4)
Show the memo (3 ticks)Memo
1. ✓
  Calculate the gradient of radius MECentre M is $(3; 0)$ .
  $m_{ME} = \frac{0 - 3}{3 - ( - 1 )}$
  $m_{ME} = - \frac{3}{4}$
  1 mark · Calculate gradient between two points
2. ✓
  Determine the gradient of tangent EC $m_{E D} \times m_{ME} = - 1$
  $m_{E D} \times (- \frac{3}{4}) = - 1$
  $∴ m_{E D} = \frac{4}{3}$
  1 mark · Use parallel and perpendicular gradients
3. ✓
  Determine the equation of tangent ECUsing point $E (- 1; 3)$ :
  $y - y_{1} = m (x - x_{1})$
  $y - 3 = \frac{4}{3} (x - (- 1))$
  $y = \frac{4}{3} x + \frac{4}{3} + 3$
  $y = \frac{4}{3} x + \frac{13}{3}$
  2 marks · Find the equation of a line
5. Question 1.2 · May/June 2024 · Paper 1 · National (DBE) · 5 marks
1.2 Solve for $x$ and $y$ simultaneously: $x + y = 9$ and $2 x^{2} - y^{2} = 7$ (5)
Show the memo (5 ticks)Memo
1. ✓
  Isolate y $y = 9 - x$
  1 mark · Change the subject of a formula
2. ✓
  Substitute into second equation $2 x^{2} - (9 - x)^{2} = 7$
  1 mark · Solve linear and non-linear simultaneous equations
3. ✓
  Simplify to standard form $x^{2} + 18 x - 88 = 0$
  1 mark · Write a quadratic in standard form
4. ✓
  Solve for both x values $x = 4$ or $x = - 22$
  1 mark · Solve a quadratic by factorising
5. ✓
  Calculate corresponding y values $y = 5$ or $y = 31$
  1 mark · Solve linear and non-linear simultaneous equations
6. Question 3.1.2 · March 2025 · Provincial · 3 marks
3.1 Given: $f (x) = \frac{2 x + 3}{- x - 3}$

Given: $f (x) = \frac{2 x + 3}{- x - 3}$
3.1.2 Hence, determine the coordinates of the intercepts of $f$ with the axes.(3)
Show the memo (3 ticks)Memo
1. ✓
  Calculate y-intercept (x=0) $y = \frac{3}{0 + 3} - 2 = 1 - 2 = - 1$
  y-intercept: $(0; - 1)$
  1 mark · Find the y-intercept
2. ✓
  Let $y = 0$ $\frac{3}{x + 3} - 2 = 0$
  1 mark · Find the x-intercepts
3. ✓
  x-intercept $\frac{3}{x + 3} = 2$
  $x = - \frac{3}{2}$ , so the x-intercept is $(- \frac{3}{2}; 0)$
  1 mark · Find the x-intercepts
7. Question 4.2 · March 2024 · Paper 1 · Provincial · 3 marks
The sketch below shows the graphs of $h (x) = - x^{2} + 2 x + p$ and $t (x) = m x + 5$ . $D$ is the $x$ -intercept, and $B$ the $y$ -intercept of $t$ . $A$ is the $y$ -intercept of $h$ . $C$ is the $x$ -intercept of $h$ . $F$ is the turning point of $h$ . $E$ is a point on $t$ , such that $EC$ is parallel to the $y$ -axis. $A B$ is 2 units and $C D$ is 7 units.
4.2 Determine the coordinates of $F$ .(3)
Show the memo (3 ticks)Memo
1. ✓
  Substitution into $x = \frac{- b}{2 a}$ $x = \frac{- b}{2 a} = \frac{- 2}{2 ( - 1 )}$
  1 mark · Find turning point of a parabola
2. ✓
  Value of x $x = 1$
  1 mark · Find turning point of a parabola
3. ✓
  Value of y $y = - (1)^{2} + 2 (1) + 3 = 4$ | $F (1; 4)$
  1 mark · Find turning point of a parabola
8. Question 5.1 · September 2024 · Paper 1 · Provincial · 2 marks
5.1 The point $P (2; 3)$ lies in the Cartesian plane. Determine the coordinates of the image of point P if P is rotated about the origin through $9 0^{0}$ in an anti-clockwise direction.(2)
Show the memo (1 tick)Memo
1. ✓
  Coordinate P' $(- 3; 2)$
  2 marks · Reflect, translate or rotate a point
9. Question 5.2 · March 2022 · Provincial · 3 marks
In the figure: $PQ$ is the diameter of the circle. $SP$ is the tangent to the circle at $P$ . $QT$ intersects the circle at $W$ and with $T$ on the line $SP$ . $QS$ intersects the circle at $R$ . $\hat{P}_{1} = x$ .
5.2 Prove $\hat{S} = x$ .(3)
Show the memo (3 ticks)Memo
1. ✓
  Calculate angle QIn $Δ PQR$ : $P \hat{Q} R = 9 0^{\circ} - x$ (Sum of angles in $Δ$ )
  1 mark · Use angle sum of a triangle
2. ✓
  Identify tangent perpendicular to diameter $S \hat{P} Q = 9 0^{\circ}$ (Diameter $⊥$ tangent)
  1 mark · Use radius and tangent facts
3. ✓
  Calculate SIn $Δ SPQ$ : $\hat{S} + 9 0^{\circ} + (9 0^{\circ} - x) = 18 0^{\circ}$
  $\hat{S} = x$
  1 mark · Use angle sum of a triangle
10. Question 7.1 · March 2022 · Provincial · 4 marks
In the diagram, $A C$ and $CE$ are tangents to the circle with centre $O$ . $B, D, G$ and $H$ are points on the circumference. $H G$ is produced to $F$ . $B D = B H$ . $\hat{D}_{1} = \hat{D}_{4} = x$ .
7.1 Find four other angles equal to $x$ .(4)
Show the memo (4 ticks)Memo
1. ✓
  Find B3 $\hat{B}_{3} = x$ (Tangents from same point $C$ , so $CB = C D$ , $Δ CB D$ is isosceles / Tan-chord theorem implies $\hat{B}_{3} = \hat{D}_{1}$ but logic usually $CB = C D$ )
  1 mark · Use isosceles triangle properties
2. ✓
  Find H1 $\hat{H}_{1} = \hat{B}_{3} = x$ (Tan-chord theorem with tangent $A C$ and chord $B D$ - Wait, $A C$ touches at $B$ )
  1 mark · Use the tan-chord theorem
3. ✓
  Find D2 $\hat{D}_{2} = \hat{H}_{1} = x$ (Angles opposite equal sides, $B D = B H$ )
  1 mark · Use isosceles triangle properties
4. ✓
  Find H2 $\hat{H}_{2} = \hat{D}_{4} = x$ (Tan-chord theorem with tangent $CE$ and chord $D G$ - wait, tangent is at $D$ , chord $D G$ subtends $H_{2}$ . Correct)
  1 mark · Use the tan-chord theorem

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