IMAT Mathematics · Geometry

Equations of Circles and Parabolas

A circle is every point at a fixed distance from its centre, which gives its equation; completing the square turns a messy equation into a centre and a radius, or a parabola into its vertex.

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Why this matters

Circle equations are the ministry papers' favourite geometry item: 2023 asked how close a circle comes to the coordinate axes, 2026 asked which statement about a given circle is true, and a 2013 question matched a centre and radius to an equation.

Concept 1 of 2: Equation of a circle and completing the square

A point

(x, y)

is on the circle when its distance from the centre is the radius. Write that distance with Pythagoras, square both sides, and you have the equation. Expanded, the equation hides the centre; completing the square brings it back.

Definition

Centre-radius form: the circle with centre $(a, b)$ and radius $r$ is $(x - a)^2 + (y - b)^2 = r^2$ .
Expanded form: $x^2 + y^2 + Dx + Ey + F = 0$ , with the same coefficient on $x^2$ and $y^2$ (divide through first if it is not 1). Then the centre is $\left(-\tfrac{D}{2}, -\tfrac{E}{2}\right)$ and $r^2 = \tfrac{D^2}{4} + \tfrac{E^2}{4} - F$ .
Completing the square: $x^2 + Dx = \left(x + \tfrac{D}{2}\right)^2 - \tfrac{D^2}{4}$ .
If $F = 0$ , the point $(0, 0)$ satisfies the equation, so the circle passes through the origin.
The centre $(a, b)$ is $|b|$ from the x-axis and $|a|$ from the y-axis. The circle cuts an axis if that distance is less than $r$ , touches it if equal, and otherwise misses it by the distance minus $r$ .

Circle with centre (a, b) and radius r

(x - a)^2 + (y - b)^2 = r^2

$(a, b)$ the centre
$r$ the radius

Worked example

Find the centre and radius of

x^2 + y^2 + 8x - 10y + 32 = 0

, and decide whether the circle meets either coordinate axis.

Practice this conceptself-check · 4 quick reps

(x minus a) squared means the centre is at plus a

(x + 4)^2 + (y - 5)^2 = 9

the centre is

(-4, 5)

, not

(4, -5)

. And the right-hand side is

r^2

: here the radius is 3, not 9.

Divide through before reading off the centre

The rule centre

= (-D/2, -E/2)

only works when

x^2

and

y^2

have coefficient 1. If the equation starts

2x^2 + 2y^2

, divide every term by 2 first.

Concept 2 of 2: The parabola y = ax² + bx + c: vertex, axis and where a line meets it

Completing the square writes a parabola as a squared bracket plus a number. The squared bracket is never negative, so the vertex is where it is zero. To see where a line meets the parabola, set the two y-expressions equal: you get a quadratic, and its number of roots is the number of meeting points.

Definition

Vertex form: $y = a(x - p)^2 + q$ has vertex $(p, q)$ and axis of symmetry $x = p$ . Equivalently $p = -\tfrac{b}{2a}$ .
$a > 0$ : opens upwards, vertex is the lowest point. $a < 0$ : opens downwards, vertex is the highest point.
The y-intercept is $c$ .
A line $y = mx + k$ meets the parabola where $ax^2 + bx + c = mx + k$ . The discriminant of this quadratic decides: positive, two points; zero, the line touches (is a tangent); negative, no meeting point.
A horizontal line $y = k$ touches an upward parabola only at the height of the vertex.

Vertex form

y = a(x - p)^2 + q \qquad \text{vertex } (p, q), \quad \text{axis } x = p

Worked example

Find the vertex of

y = x^2 - 4x + 1

, and find where the line

y = 2x - 8

meets this parabola.

Practice this conceptself-check · 4 quick reps

Touching means a repeated root

A line touches a parabola when the combined quadratic has exactly one solution, which means a discriminant of zero. Two different roots mean the line cuts through at two points; no real roots mean it misses.

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Equation of a circle and completing the square
Circle with centre (a, b) and radius r
$(x - a)^2 + (y - b)^2 = r^2$
The parabola y = ax² + bx + c: vertex, axis and where a line meets it
Vertex form
$y = a(x - p)^2 + q \qquad \text{vertex } (p, q), \quad \text{axis } x = p$

Watch out for (3)

(x minus a) squared means the centre is at plus a
→ Equation of a circle and completing the square
Divide through before reading off the centre
→ Equation of a circle and completing the square
Touching means a repeated root
→ The parabola y = ax² + bx + c: vertex, axis and where a line meets it

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