Equation of a circleEdexcel International A Level Maths: Revision notes

A circle is the set of points a fixed distance $r$ (the radius) from a fixed point $(a,b)$ (the centre). Using the distance formula, a point $(x,y)$ is on the circle when $\sqrt{(x-a)^2+(y-b)^2}=r$, which gives $(x-a)^2+(y-b)^2=r^2.$ If the centre is the origin, this becomes $x^2+y^2=r^2$.

The right-hand side is $r^2$, so the radius is its square root.

From $(x-a)^2+(y-b)^2=r^2$, read off the centre $(a,b)$ and radius $r$. The signs in the brackets are the opposite of the coordinates.

Example: $(x-3)^2+(y+2)^2=25$. Write $(y+2)=(y-(-2))$, so the centre is $(3,-2)$ and the radius is $\sqrt{25}=5$.

Example: $(x+4)^2+(y-1)^2=13$. The centre is $(-4,1)$ and the radius is $\sqrt{13}$.

To write the equation you need the centre $(a,b)$ and $r^2$.

Centre and a point on the circle. The radius is the distance from the centre to the point. Centre $(-4,1)$ through $(-2,4)$: $r^2=(-2+4)^2+(4-1)^2=4+9=13$, so $(x+4)^2+(y-1)^2=13$.

Diameter given by its end points. The centre is the midpoint of the diameter, and $r$ is half the length of the diameter. For $A(-3,1)$ and $B(5,5)$ the centre is $(1,3)$ and $r^2=(5-1)^2+(5-3)^2=20$, so $(x-1)^2+(y-3)^2=20$.

A circle may be given as $x^2+y^2+px+qy+c=0$. Complete the square in $x$ and in $y$ to find the centre and radius.

Example: $x^2+y^2-10x+4y+13=0$. $(x-5)^2-25+(y+2)^2-4+13=0$, so $(x-5)^2+(y+2)^2=16$. The centre is $(5,-2)$ and the radius is $4$.

If the number on the right-hand side is not positive, there is no circle.

To find where a circle meets a line, substitute the line's equation into the circle's equation and solve.

For the circle above, the $x$-axis is $y=0$: $(x-5)^2+4=16$, so $(x-5)^2=12$ and $x=5\pm2\sqrt3$.

To test a point $(p,q)$, calculate $(p-a)^2+(q-b)^2$ and compare with $r^2$. If it is less than $r^2$ the point is inside the circle, if equal it is on the circle, and if greater it is outside.

Example: for $(x-3)^2+(y+2)^2=25$ the point $(6,2)$ gives $9+16=25$, so it lies on the circle.