# Equation of a Circle

## Equation of a Circle Notes

We find the **equation of a circle** from the coordinates of its centre and its radius. Consider this circle whose **centre is at the point (a,b) **and whose **radius is r**. The equation of the circle is given by

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

The values of $a$, $b$ and $r$ are specified whereas $x$ and $y$ are left as general points. This is much like when we specify $m$ and $c$ in $y=mx+c$ when we have the equation of a given straight line. Note that the equation of a straight line is *explicit – *y is the subject and is given in terms of x. The circle equation, on the other hand, is implicit – the $x$s and $y$s are mixed up.

### Things to Learn for Circle Equation Questions

**You should know the following facts:**

- Most importantly, the equation of a circle.
- Secondly, that the angle subtended from a diameter at the circumference is a right angle.
- Next, the radius and tangent touch at right angles to one another.
- Finally, a perpendicular from a chord bisects the chord.

**You should be able to find the equation of a circle:**

- using completing the square and/or Pythagoras,
- given a triangle that has all three points on its circumference and
- find the equation of a tangent using perpendicular gradients.