Coordinate Geometry of the Circle (Leaving Cert Higher Level)

Coordinate geometry of the circle covers the equation of a circle, finding its centre and radius, tangents and intersections with lines. It is examined on Paper 2.

Written by Sean Doyle · Last updated: 2026-06-12

Key facts

A circle centred at the origin has equation x² + y² = r².
The general circle equation gives the centre and radius after completing the square.
A tangent touches the circle at exactly one point and is perpendicular to the radius there.
Coordinate geometry of the circle is examined on Paper 2.

Coordinate Geometry of the Circle explained

What is Coordinate Geometry of the Circle?

Coordinate geometry of the circle is the study of circles using the Cartesian plane.

We describe circles using equations rather than geometric constructions.

This allows us to: - Find exact points of intersection - Calculate tangent lines algebraically - Solve problems using substitution and algebra

Circle Terminology

Key terms you need to know:

Centre: The fixed point equidistant from all points on the circle
Radius ( $r$ ): The distance from the centre to any point on the circle
Diameter ( $d$ ): A line through the centre, where $d = 2r$
Chord: A line segment with both endpoints on the circle
Tangent: A line that touches the circle at exactly one point

The Basic Equation

A circle with centre at the origin $(0, 0)$ and radius $r$ has equation:

$x^2 + y^2 = r^2$

Every point $(x, y)$ on the circle satisfies this equation.

This comes from the Pythagorean theorem — the distance from any point $(x, y)$ to the origin is $r$ .

This is an introduction. The full Coordinate Geometry of the Circle lessons, worked solutions and practice questions are available to members. Start your free trial.

Key formulas

Name	Formula	Description
Circle Equation (Centre-Radius)	$(x-h)^2 + (y-k)^2 = r^2$	Circle with centre (h,k) and radius r
Circle Equation (General)	$x^2 + y^2 + 2gx + 2fy + c = 0$	General form of circle equation
Centre from General Form	$\text{Centre} = (-g, -f)$	Centre of circle from general form
Radius from General Form	$r = \sqrt{g^2 + f^2 - c}$	Radius of circle from general form
Tangent at Point	$(x-h)(x_1-h) + (y-k)(y_1-k) = r^2$	Tangent to circle at point (x₁,y₁)
Length of Tangent	$L = \sqrt{(x_1-h)^2 + (y_1-k)^2 - r^2}$	Length of tangent from external point to circle

Worked examples

Worked example 1

What is the equation of a circle with centre $(0, 0)$ and radius $7$ ?

For a circle at the origin with radius $r$ , the equation is $x^2 + y^2 = r^2$ . Here $r = 7$ , so $r^2 = 49$ .

Answer:

x² + y² = 49

Worked example 2

Find the centre of the circle $x^2 + y^2 - 6x + 4y - 12 = 0$ .

From general form $x^2 + y^2 + 2gx + 2fy + c = 0$ , centre is $(-g, -f)$ . Here $2g = -6$ so $g = -3$ , and $2f = 4$ so $f = 2$ . Centre = $(-(-3), -2) = (3, -2)$ .

Answer:

(3, -2)

Where students lose marks

Sign errors reading the centre from the circle equation.
Forgetting the radius is squared in the equation.
Not using the perpendicular-radius property for tangent questions.

Frequently asked questions

What is the equation of a circle?

A circle centred at (h, k) with radius r has equation (x - h)² + (y - k)² = r². Centred at the origin it simplifies to x² + y² = r².

How do I find where a line meets a circle?

Substitute the line's equation into the circle's equation and solve the resulting quadratic; the number of solutions tells you how many intersection points there are.

Is coordinate geometry of the circle on Paper 2?

Yes, it is examined on Paper 2.

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