# 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.

The derivation of the equation of a circle is from an application of Pythagoras Theorem. Specifically, the longest side squared is equal to the sum of the squares of the shorter sides. Evidently, the longest side is the radius. If we pick a point on the circle $(x,y)$, we see that the shorter sides have lengths $x-a$ and $y-b$. It follows that $(x-a)^2+(y-b)^2=r^2$.

They may present you with questions that expect you to know the equation of a circle. Other questions might bring in knowledge from other areas of maths such as finding mid-points. In more complicated questions they may ask you to find gradients using the knowledge that a tangent to a circle is perpendicular to its radius.

### Things to Learn for Circle Equation Questions

You should know the following facts:

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

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

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

## Examples

1. The centre of a circle is given by (2,-5) and its radius is the square root of 11. Write down the equation of the circle.
2. Find the equation of the circle given that the centre is at (1,2) and the point (3,5) lies on the circle.
3. Find the equation of the tangent to the circle $(x-4)^2+(y-1)^2=10$ at the point (3,4).

### (Completing the Square)

The equation of a circle is given by
$x^2+6x+y^2-4x+9=0$.

1. Find the centre and the radius of the circle.
2. Find all points on the circle with x-coordinate -4.

The three points $(1,-2)$, $(4,5)$ and $(5,x)$ lie on the circumference of a circle. The points $(1,-2)$ and $(4,5)$ are at either end of the diameter. Find the exact value of $x$.
The chord that joins the circumferential points $(x,-4)$ and $(-2,-2)$ bisects the diameter of the circle whose centre is at $\left(1,-\frac{5}{2}\right)$. Find the value of $x$.