Form 5 maths

Equations of circles

Q: Why is the centre (−D/2, −E/2)?

Expanding (x − h)² gives −2hx. Matching −2h = D gives h = −D/2.

Q: How do I know when to use Δ?

Whenever the question says "tangent", "touches", or "intersects at only one point", and you don't know the point of contact.

Q: What if r² comes out negative?

Then it isn't a real circle. Check your arithmetic, or the question is testing exactly that.

The standard and general forms of the equation of a circle, and (NF) where a line meets a circle and how to find tangents. It is a favourite topic of the long coordinate geometry questions.

What you will learn

13.1 The equation of a circle from given conditions, such as the centre and the radius.
13.1 The general form x² + y² + Dx + Ey + F = 0: find the centre and radius.
NF 13.2 Points of intersection of a line and a circle; the number of intersections; equations of tangents.

You need first: Coordinates, Equations of straight lines, Quadratic equations in one unknown, Loci, Basic properties of circles

Key terms

English	中文
equation of a circle	圓方程
standard form, general form	標準式，一般式
centre, radius	圓心，半徑
point of intersection	交點
tangent to a circle	圓的切線
inside / on / outside the circle	在圓內／圓上／圓外

Standard steps

Two forms

Form	Equation	Centre	Radius
standard	(x − h)² + (y − k)² = r²	(h, k)	r
general	x² + y² + Dx + Ey + F = 0	(−D/2, −E/2)	√[(D/2)² + (E/2)² − F]

Why: the standard form is the distance formula: every point (x, y) is r away from (h, k).

Standard form: read the centre and radius directly. Watch the signs: (x − 3)² + (y + 2)² = 25 has centre (3, −2).
General form: make the x² and y² coefficients 1 first, then centre (−D/2, −E/2) and the radius formula.
To change standard to general, expand and collect terms. To change general to standard, complete the square.

Point inside, on or outside

Substitute the point into the left side of the general form: negative → inside; 0 → on; positive → outside.

Line and circle (NF)

Make y (or x) the subject of the line.
Substitute into the circle. You get a quadratic in x.
Use Δ:

Δ	The line…
> 0	cuts the circle at 2 points
= 0	is a tangent (1 point)
< 0	misses the circle

To find the points, solve the quadratic and substitute back.

Tangent at a known point on the circle (NF)

Slope of the radius from the centre to the point.
Tangent slope = −1 ÷ radius slope (tangent ⊥ radius).
Point-slope form.

Worked examples

Example 1: Two forms: centre (3, −2), radius 5. Write the equation in standard form and in general form.

(x − 3)² + (y + 2)² = 25
x² − 6x + 9 + y² + 4y + 4 = 25
x² + y² − 6x + 4y − 12 = 0

Example 2: Find the centre and radius of x² + y² + 4x − 6y − 3 = 0.

Centre (−2, 3); radius √(2² + 3² + 3) = √16 = 4.

Example 3: NF: find k if y = x + k is a tangent to x² + y² = 8.

x² + (x + k)² = 8
2x² + 2kx + k² − 8 = 0
Δ = (2k)² − 4(2)(k² − 8) = −4k² + 64 = 0
k = 4 or k = −4

Example 4: Tangent at a known point on the circle (NF): find the tangent to x² + y² − 6x + 4y − 12 = 0 at (6, 2).

centre (3, −2); slope of radius = (2 + 2)/(6 − 3) = 4/3
tangent slope = −3/4
y − 2 = −¾(x − 6)
3x + 4y − 26 = 0

Common mistakes

Centre sign: (x − 3)² + (y + 2)² has centre (3, −2).
Radius from the general form without the square root, or with F's sign wrong.
Forgetting the equation must have x² and y² with the same coefficient; divide through first if it's 2x² + 2y² + ….
Using the line's slope as the tangent slope at a point.

How the DSE tests it

Section A: equation from the centre and a point, or centre and radius from the general form; point inside/outside.
Section B (NF): the long coordinate question. Find a circle (often a circumcircle or a locus), find a tangent by Δ = 0, then a distance, angle or area, sometimes with the triangle centres from F3.

Questions you might not ask in class

Why is the centre (−D/2, −E/2)?