Grade 12 Maths · Analytical Geometry · Decide if a point is inside a circle

Decide if a point is inside a circle Grade 12

Decide whether a point lies inside, on or outside a circle by comparing its distance from the centre with the radius, for example (4; 3) is 5 from the origin, so it lies on x^2 + y^2 = 25.

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Grade 12: The relationship between a circle and a pointMrs Le Roux - Maths

How to Prove that a Point Lies on a Circle (or Inside/Outside)The Complete Guide to Everything

Points inside/outside/on a circle | Mathematics I | High School Math | Khan AcademyKhan Academy

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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)

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More questions: decide if a point is inside a circle

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3. Question 4.1.4 · September 2024 · Paper 2 · Provincial · 3 marks
4.1 The diagram below shows two circles touching at $J (2; - 1)$ .
The smaller circle has its centre at the origin and the bigger circle has centre $K (a; - 3)$ .
The length of the radius of the bigger circle is TWICE the length of the radius of the smaller circle.
SR is a tangent to both circles, touching the bigger circle at S and the smaller circle at R.
KO and SR are both produced to intersect in point P.

Answer the questions that follow. $Diagram for question 4.1$
4.1.4 Does the point $(10; - 4)$ lie outside, inside or on the bigger circle?(3)
Show the memo (3 ticks)Memo
1. ✓
  SubstitutionSubstitute $(10; - 4)$ :
  $(10 - 6)^{2} + (- 4 + 3)^{2}$
  1 mark · Use the equation of a circle
2. ✓
  Comparison $= 17$
  $17 < 20$
  1 mark · Decide if a point is inside a circle
3. ✓
  Conclusion $∴$ the point lies inside the circle
  1 mark · Decide if a point is inside a circle
4. Practice · NavyBlue · 2 marks
A circle has equation $x^{2} + y^{2} - 4 x + 6 y - 12 = 0$ .
(b) Show that $P (6; 0)$ lies on the circle and that $Q (0; 1)$ lies inside it.(2)
Show the memo (2 ticks)Memo
1. ✓
  P $M P^{2} = 4^{2} + 3^{2} = 25 = r^{2}$ , so $P$ is on the circle
  1 mark · Decide if a point is inside a circle
2. ✓
  Q $M Q^{2} = (- 2)^{2} + 4^{2} = 20 < 25$ , so $Q$ is inside the circle
  1 mark · Decide if a point is inside a circle
5. Practice · NavyBlue · 2 marks
Circle $C_{1}$ has equation $x^{2} + y^{2} = 20$ and centre $O$ . Circle $C_{2}$ has equation $x^{2} + y^{2} - 12 x - 6 y + 40 = 0$ and centre $K$ .
Determine whether the point $Q (1; 4)$ lies inside, on or outside each circle.(2)
Show the memo (2 ticks)Memo
1. ✓
  Relative to C1 $O Q^{2} = 1^{2} + 4^{2} = 17 < 20$ , so $Q$ lies inside $C_{1}$ .
  1 mark · Decide if a point is inside a circle
2. ✓
  Relative to C2 $K Q^{2} = (1 - 6)^{2} + (4 - 3)^{2} = 26 > 5$ , so $Q$ lies outside $C_{2}$ .
  1 mark · Decide if a point is inside a circle
6. Practice · NavyBlue · 3 marks
For which values of $p$ does the point $(p; 2)$ lie inside the circle $x^{2} + y^{2} = 13$ ?(3)
Show the memo (3 ticks)Memo
1. ✓
  Inside condition $p^{2} + 2^{2} < 13$
  1 mark · Decide if a point is inside a circle
2. ✓
  Simplify $p^{2} < 9$
  1 mark · Decide if a point is inside a circle
3. ✓
  Interval $- 3 < p < 3$
  1 mark · Solve a quadratic inequality

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