Equations of Circles

Standard form

A circle with centre $(h, k)$ and radius $r$:

$$(x - h)^2 + (y - k)^2 = r^2$$

General form

$$x^2 + y^2 + Dx + Ey + F = 0$$ $$\text{centre} = \left(-\frac{D}{2}, -\frac{E}{2}\right), \qquad r = \sqrt{\left(\frac{D}{2}\right)^2 + \left(\frac{E}{2}\right)^2 - F}$$

Example: $x^2 + y^2 - 6x + 4y - 12 = 0$ has centre $(3, -2)$ and radius $\sqrt{9 + 4 + 12} = 5$.

Common mistake: the general form only works when the coefficients of $x^2$ and $y^2$ are both 1. Divide through first if they are, say, 2.

Position of a point

Substitute the point into $x^2 + y^2 + Dx + Ey + F$:

• negative: the point is inside the circle,
• zero: on the circle,
• positive: outside the circle.

A line and a circle

Substitute the line into the circle to get a quadratic, then use $\Delta$:

$\Delta$	Meaning
$\Delta > 0$	two points of intersection
$\Delta = 0$	the line is a tangent
$\Delta < 0$	no intersection

You can also compare the distance from the centre to the line with $r$.

The four centres of a triangle

Centre	Lines that meet there	Property
Circumcentre	Perpendicular bisectors of the sides	Equidistant from the three vertices; centre of the circumscribed circle
In-centre	Angle bisectors	Equidistant from the three sides; centre of the inscribed circle
Orthocentre	Altitudes
Centroid	Medians	Coordinates are the mean of the vertices' coordinates

For a right-angled triangle, the circumcentre is the mid-point of the hypotenuse.

Tangent at a point on the circle

The tangent is perpendicular to the radius. Find the slope of the radius from the centre to the point, take $-\dfrac{1}{m}$, then use point-slope form.

Exam tip: many questions combine this topic with loci. "The locus of a point that is always 5 units from $(3, -2)$" is a circle: $(x - 3)^2 + (y + 2)^2 = 25$.