MathsISC Class 11 · Circles

Circle through Given Points, Intercepts on the Axes and Relative Position of Two Circles

Q: What if the three points are collinear?

No circle passes through three collinear points; the system for g, f, c has no solution.

Q: How many circles pass through two points?

Infinitely many; a third condition is needed.

Q: What is the radical axis?

The line through the points of intersection of two circles; it is outside this syllabus unit.

Updated 17 October 2026 · Fact-checked

A circle through three non-collinear points is found by substituting them into x² + y² + 2gx + 2fy + c = 0 and solving for g, f, c. The intercepts on the axes are 2√(g² − c) and 2√(f² − c). Two circles with centres at distance d and radii r₁, r₂ touch externally if d = r₁ + r₂, touch internally if d = |r₁ − r₂|, intersect if |r₁ − r₂| < d < r₁ + r₂.

Compare the distance d between centres with r₁ + r₂ and |r₁ − r₂| to decide how two circles meet.

Illustration drawn for QuestionBuddy.

Understand Circles through Points, Intercepts and Two Circles

Circle through three points. Substitute the three points into $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ to get three linear equations in g, f, c. A point at the origin gives c = 0 at once. Alternatively, the centre lies on the perpendicular bisectors of the chords.

Centre on a line, or two points given. If the circle passes through two points A and B and its centre lies on a given line, the centre is at the intersection of that line with the perpendicular bisector of AB. Equating the squared distances from the centre to A and B is a quick way to find the centre.

Circles touching the axes. A circle of radius r touching both axes in the first quadrant has centre (r, r) and equation $(x - r)^{2} + (y - r)^{2} = r^{2}$ . A circle touching the x-axis has $∣ k ∣ = r$ ; touching the y-axis has $∣ h ∣ = r$ .

Intercepts. Putting y = 0 in the general form gives $x^{2} + 2 g x + c = 0$ with the x-intercept length $2 g^{2} - c$ . Similarly, the y-intercept has length $2 f^{2} - c$ . The circle touches the x-axis if $g^{2} = c$ , and the y-axis if $f^{2} = c$ .

Two circles. Let the centres be $C_{1}, C_{2}$ with $d = C_{1} C_{2}$ and radii $r_{1}, r_{2}$ :

$d > r_{1} + r_{2}$ : separate; no common point.
$d = r_{1} + r_{2}$ : touch externally.
$∣ r_{1} - r_{2} ∣ < d < r_{1} + r_{2}$ : intersect at two points.
$d = ∣ r_{1} - r_{2} ∣$ : touch internally.
$d < ∣ r_{1} - r_{2} ∣$ : one lies inside the other.

Circles through Points, Intercepts and Two Circles: formulas to remember

Through three points: $Substitute into $x^2 + y^2 + 2gx + 2fy + c = 0$ and solve$; Point (0, 0) gives c = 0.
Intercepts: $x-intercept: $2\sqrt{g^2 - c}$ | y-intercept: $2\sqrt{f^2 - c}$$; Touches the axis if the root is 0.
Touching both axes: $Centre $(r, r)$, equation $(x - r)^2 + (y - r)^2 = r^2$$; First quadrant.
Two circles: $External: $d = r_1 + r_2$ | Internal: $d = |r_1 - r_2|$ | Two points: between them$; d = distance between centres.

Circles through Points, Intercepts and Two Circles in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Planning a circular lawn through three posts

Suppose three posts stand at (0, 0), (4, 0) and (0, 6) metres, and a circular path is to pass through all of them.

Find the equation of the circle, its centre and radius.

Try it first, then show the solutionSolution

Through the origin: $c = 0$ . (4, 0): $16 + 8 g = 0$ gives $g = - 2$ . (0, 6): $36 + 12 f = 0$ gives $f = - 3$ .
Equation: $x^{2} + y^{2} - 4 x - 6 y = 0$ .
Centre $(2, 3)$ , radius $4 + 9 = 13 \approx 3.61$ m.

Answer: x² + y² − 4x − 6y = 0; centre (2, 3); r = √13.

Memory hook: A right angle at the origin means the hypotenuse is a diameter.

Tricks to remember Circles through Points, Intercepts and Two Circles

Short memory aids. Say each one aloud once, then use it on a question the same day.

Origin gives c = 0
If the circle passes through the origin, drop c.
Use it when: Fewer unknowns.
Touch means zero root
Touching the x-axis: g² = c.
Use it when: Intercept conditions.

Circles through Points, Intercepts and Two Circles: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Circle touching both axes · 1 mark · MCQBoard level

A circle of radius 3 touches both axes and lies in the first quadrant. Its centre is:
(a) (3, 3)
(b) (0, 3)
(c) (3, 0)
(d) (−3, 3)

Try it first, then show the solutionSolution

The centre is at distance 3 from each axis, in the first quadrant: (3, 3).

Answer: (a) (3, 3)

Example 2: Circle through three points · 3 marks · Short answerBoard level

Find the equation of the circle through (0, 0), (4, 0) and (0, 6), and its centre.

Try it first, then show the solutionSolution

Through the origin: $c = 0$ . (0.5 mark)
(4, 0): $16 + 8 g = 0$ , so $g = - 2$ ; (0, 6): $36 + 12 f = 0$ , so $f = - 3$ . (1.5 marks)
Equation $x^{2} + y^{2} - 4 x - 6 y = 0$ ; centre (2, 3). (1 mark)

Answer: x² + y² − 4x − 6y = 0; centre (2, 3).

Example 3: Intercepts · 3 marks · Short answerHarder board level

Find the lengths of the intercepts made by the circle $x^{2} + y^{2} - 6 x + 4 y - 12 = 0$ on the x-axis and y-axis. Does it touch either axis?

Try it first, then show the solutionSolution

$g = - 3$ , $f = 2$ , $c = - 12$ . (0.5 mark)
x-intercept: $2 g^{2} - c = 2 9 + 12 = 221 \approx 9.17$ . (1 mark)
y-intercept: $2 f^{2} - c = 2 4 + 12 = 8$ . (1 mark)
Neither $g^{2} = c$ nor $f^{2} = c$ , so it touches neither axis. (0.5 mark)

Answer: x-intercept 2√21; y-intercept 8; touches neither axis.

Example 4: Relative position of two circles · 4 marks · Short answerHarder board level

Determine the position of the circles (i) $x^{2} + y^{2} = 25$ and $(x - 7)^{2} + y^{2} = 4$ , and (ii) $x^{2} + y^{2} - 6 x - 8 y = 0$ and $x^{2} + y^{2} = 25$ .

Try it first, then show the solutionSolution

(i) Centres (0, 0) and (7, 0): $d = 7$ ; $r_{1} = 5$ , $r_{2} = 2$ ; $d = r_{1} + r_{2}$ : touch externally. (2 marks)
(ii) Second circle: centre (3, 4), $r_{2} = 9 + 16 = 5$ ; $d = 5$ from (0, 0). (1 mark)
$r_{1} - r_{2} = 0 < d = 5 < 10$ : the circles intersect at two points. (1 mark)

Answer: (i) external contact; (ii) two points of intersection.

Example 5: Centre on a given line · 5 marks · Long answerHOTS / competency

Find the equation of the circle passing through (1, 2) and (3, 4) with its centre on the x-axis. Find its radius and check that both points lie on it.

Try it first, then show the solutionSolution

Let the centre be $(h, 0)$ . Equal distances: $(h - 1)^{2} + 4 = (h - 3)^{2} + 16$ . (1.5 marks)
$h^{2} - 2 h + 5 = h^{2} - 6 h + 25$ , so $4 h = 20$ and $h = 5$ . (1.5 marks)
$r^{2} = (5 - 1)^{2} + 4 = 20$ ; equation $(x - 5)^{2} + y^{2} = 20$ . (1 mark)
Check (3, 4): $4 + 16 = 20$ ✓; (1, 2): $16 + 4 = 20$ ✓. (1 mark)

Answer: (x − 5)² + y² = 20; radius 2√5.

Where students lose marks in Circles through Points, Intercepts and Two Circles

The slips examiners see most often, and what to write instead.

Mistake: Wrong condition for touching circles
Using r₁ + r₂ for internal contact.
Do this: Internal: d = |r₁ − r₂|; external: d = r₁ + r₂.
Usually costs: 2 marks.
Mistake: Sign slip in solving for g and f
Dropping the minus sign.
Do this: Solve the linear equations carefully and verify with the points.
Usually costs: 1 to 2 marks.
Mistake: Intercept formula without the factor 2
Giving √(g² − c).
Do this: The intercept length is 2√(g² − c).
Usually costs: 1 mark.

Exam tips

Check that each given point satisfies your equation.
Use the origin to set c = 0 quickly.
Compare d with r₁ + r₂ and |r₁ − r₂| in an order.
Sketch the circles to confirm the result.

Practice questions on Circles through Points, Intercepts and Two Circles

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
Two circles with radii 3 and 5 whose centres are 8 apart:
(a) touch externally
(b) touch internally
(c) intersect at two points
(d) do not meet and are separate
Hint
Compare d with r₁ + r₂.
Answer
(a) touch externally
Q2 · 2 marks · Short answerBoard level
A circle has its centre at (5, −4) and touches the y-axis. Write its equation.
Hint
Radius equals |h|.
Answer
Radius $= 5$ : $(x - 5)^{2} + (y + 4)^{2} = 25$ .
Q3 · 3 marks · Short answerHarder board level
Find the circle through (1, 0), (0, 1) and (−1, 0).
Hint
Use g, f, c.
Answer
Points give $1 + 2 g + c = 0$ , $1 + 2 f + c = 0$ , $1 - 2 g + c = 0$ : $g = 0$ , $c = - 1$ , $f = 0$ : $x^{2} + y^{2} = 1$ .
Q4 · 3 marks · Short answerHarder board level
Show that $x^{2} + y^{2} - 8 x - 6 y + 16 = 0$ touches the x-axis and find the point of contact.
Hint
Check g² = c.
Answer
$g = - 4$ , $c = 16 = g^{2}$ : it touches the x-axis. Centre $(4, 3)$ , radius $16 + 9 - 16 = 3$ ; point of contact $(4, 0)$ .
Q5 · 3 marks · Short answerHarder board level
Find the equation of the circle with radius 5 that touches the y-axis at (0, 3) and lies in the first quadrant.
Hint
The centre is on y = 3.
Answer
Centre $(5, 3)$ : $(x - 5)^{2} + (y - 3)^{2} = 25$ .
Q6 · 4 marks · Case-basedHOTS / competency
Case study: Suppose two circular flower beds have equations $x^{2} + y^{2} = 16$ and $(x - 3)^{2} + y^{2} = 1$ (metres).
(i) State the centres and radii. (1 mark)
(ii) Find the distance between the centres. (1 mark)
(iii) Find the relative position. (1 mark)
(iv) State whether a path along the x-axis crosses the second bed. (1 mark)
Hint
Compare d with |r₁ − r₂|.
Answer
(i) (0, 0), 4 and (3, 0), 1. (ii) $d = 3$ . (iii) $d = r_{1} - r_{2} = 3$ : touch internally (the small bed is inside, touching at (4, 0)). (iv) Yes: it crosses the small bed from x = 2 to x = 4.

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.x-intercept length?Show answer
2√(g² − c).
2.External contact?Show answer
d = r₁ + r₂.
3.Circle touches x-axis when?Show answer
g² = c.
4.Centre of a circle touching both axes?Show answer
(r, r) in the first quadrant.

Circles through Points, Intercepts and Two Circles: frequently asked questions

What if the three points are collinear?

No circle passes through three collinear points; the system for g, f, c has no solution.

How many circles pass through two points?

Infinitely many; a third condition is needed.

What is the radical axis?

The line through the points of intersection of two circles; it is outside this syllabus unit.