Introduction

Video lesson3 worked examples

The equation of a circle with centre $(0,0)$ is $x^2+y^2=r^2$

Radius or diameter? the equation uses $r^2$ , so a question that gives the diameter needs halving first. This is the most common single error in circle work.
The angle in a semicircle is a right angle: it is worth knowing, because it converts many circle problems into perpendicular-gradient problems.
A tangent is perpendicular to the radius at the point of contact: this is the fact almost every tangent question depends on.
Keep the radius exact: leave it as a surd rather than rounding — a rounded radius will not satisfy the equation later.

Example 1

State the equation of the circle with centre $(0,0)$ and radius 5 units.

x^2 + y^2 = r^2

x^2 + y^2 = 5^2

x^2 + y^2 = 25

Example 2

State the centre and radius of the circle with equation $x^2+y^2=121$ .

Centre is $(0,0)$ .

r^2 = 121

r = \sqrt{121} = 11

Radius is 11 units.

Example 3

State the equation of the circle with centre $(0,0)$ which passes through the point $(-5,12)$ .

Substitute $x = -5$ and $y = 12$ into $x^2 + y^2 = r^2$ to find $r^2$ :

(-5)^2 + 12^2 = r^2

25 + 144 = r^2

169 = r^2

The equation is:

x^2 + y^2 = 169