Circles Worked Solutions — A-Level Maths

Recall the standard equation of a circle

Every circle can be written this way, where $(a,b)$ is the centre and $r$ is the radius. Our job is just to slot the right numbers into $a$, $b$ and $r$.

Write down the centre and radius

The centre gives us $a=3$ and $b=2$, and the radius is $r=5$. Keep them separate so we do not mix up the two brackets.

Substitute into the formula

Replace $a$ with $3$ and $b$ with $2$. The right-hand side is the radius squared, so we write $5^2$ for now.

Work out the radius squared

Squaring the radius gives $5^2=25$. That finishes the equation of the circle.

Recall the equation of a circle at the origin

When the centre is the origin $(0,0)$ the brackets $(x-0)$ and $(y-0)$ are just $x$ and $y$. So the equation simplifies to $x^2+y^2=r^2$.

Insert the radius

The radius is $7$, so the right-hand side is $7^2$. Nothing else changes.

Square the radius

Since $7^2=49$, the equation of the circle is $x^2+y^2=49$.

Compare with the standard form

We match our equation to the general form. The numbers being subtracted inside the brackets give the centre $(a,b)$.

Read off a and b

Inside the brackets we subtract $4$ from $x$ and $1$ from $y$, so $a=4$ and $b=1$. Be careful: it is the number that is subtracted.

State the centre

The centre of the circle is therefore $(4,\ 1)$.

Compare with the standard form

The right-hand side of the standard equation is the radius squared. So whatever number is on the right equals $r^2$.

Identify r squared

Here the right-hand side is $49$, so $r^2=49$.

Take the square root

The radius is a length so we take the positive square root: $\sqrt{49}=7$.

Set the right-hand side equal to r squared

The number on the right of the equation is the radius squared, so $r^2=20$.

Take the square root

Taking the positive square root gives $r=\sqrt{20}$. This is not a whole number, so we simplify the surd.

Simplify the surd

We look for the biggest square factor of $20$, which is $4$. Since $\sqrt{4}=2$ we get $2\sqrt{5}$. This links back to surds work from the indices and surds topic.