Equation of a Circle (Cambridge (CIE) IGCSE Additional Maths): Revision Note

Equation of a circle

What is the equation of a circle?

• A circle with centre (a, b) and radius r has the equation

${\mathbf{(}\mathit{x}\mathbf{-}\mathit{a}\mathbf{)}}^{\mathbf{2}}\mathbf{+}{\mathbf{(}\mathit{y}\mathbf{-}\mathit{b}\mathbf{)}}^{\mathbf{2}}\mathbf{=}{\mathit{r}}^2$ 

•  You need to be able to find the equation of a circle given its centre and radius

  • Substitute the values into the formula

How do I find the centre and radius of a centre given its equation?

• Make sure it is in the form ${\left(x-a\right)}^2+{\left(y-b\right)}^2=r^2$

  • The radius is the positive square root of the constant term

  • The coordinates of the centre can be found by finding the values that make each bracket equal to zero

Finding the centre & radius

What are the different forms of the equation of a circle?

• The most useful equation of a circle is ${\left(x-a\right)}^2+{\left(y-b\right)}^2=r^2$

  • This is so the centre, $\left(a,b\right)$ and radius $r$ are easy to see

• Any other form of the equation of a circle can be rearranged into this form

  • The most common alternative form for the equation of a circle is called the general form $x^2+y^2+2gx+2fy+c=0$

How do I find the centre and radius of a circle from any form of its equation?

• A circle equation in a different form can always be rearranged into (x- a)² + (y - b)² = r²

  • The centre is then $\left(a,b\right)$ and radius $r$

• Rearranging to this form will often involve completing the square

Intersection of a circle & a line

What is meant by the intersection of a circle and a line?

• A line may pass through a circle

  • in which case it will intersect the circle twice

  • the part of the line between the two points of intersection will be a chord

    • or, if it passes through the centre of the circle, a diameter

• A line may touch a circle

  • in which case it will intersect the circle once

  • such a line would be called a tangent to the circle

• A line may not intersect a circle at all

How do I determine whether a line and a circle intersect?

• For the equation of a circle in the form ${\left(x-a\right)}^2+{\left(y-b\right)}^2=r^2$ and the equation of a line in the form $y=mx+c$

• STEP 1
  Substitute the linear equation into the circle equation

  • e.g. ${\left(x-5\right)}^2+{\left(y-2\right)}^2=13$ and $y=x-4$ would become ${\left(x-5\right)}^2+{\left(\left(x-4\right)-2\right)}^2=13$

• STEP 2
  Expand, rearrange and simplify this equation - it should be a quadratic

  • e.g. ${\left(x-5\right)}^2+{\left(x-6\right)}^2=13$ $x^2-10x+25+x^2-12x+36-13=0$ $x^2-11x+24=0$

• STEP 3
  Solve the equation to deduce the number of intersections
  If there are two solutions, there are two intersections, one solution (repeated) indicates a tangent, no (real) solutions indicates no intersection

  • e.g.   $$\begin{gathered} \left(x-3\right)\left(x-8\right)=0 \\ x=3,x=8 \end{gathered}$$ Two solutions so the line and the circle intersect twice

• STEP 4 If required, find the $y$-coordinates of the intersection(s)

  • e.g.

$$\begin{gathered} y=x-4 \\ y=3-4=-1,y=8-4=4 \end{gathered}$$

The line and the circle intersect at the points $\left(3,-1\right)$ and $\left(8,4\right)$