Intersection of Two Circles

What is meant by the intersection of two circles?

Two circles may intersect once (touch), twice (cross), or not at all
- Touching circles may be referred to as tangent to each other
  - they would have a common tangent line

How do I determine if two circles intersect or not?

Find the distance, $d$ , between the centres of the two circles
- This can be found using Pythagoras' theorem
  - For centres $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ , $d^{2} = {(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}$
The radii of the two circles, $r_{1}$ and $r_{2}$ , where $r_{2} \geq r_{1}$ are also needed
- If $r_{2} - r_{1} < d < r_{1} + r_{2}$ then the circles intersect twice

- If $d = r_{2} - r_{1}$ or $d = r_{1} + r_{2}$ then the circles intersect once

- If $d > r_{1} + r_{2}$ or $d < r_{2} - r_{1}$ then the circles do not intersect

The different cases for two circles intersecting

Rather than trying to remember those formulae, try to understand the logic behind each situation

How do I find the coordinates of the point(s) of intersection of two circles?

Once it has been determined that the circles do intersect at least once, the following process can be used to determine the coordinates of any intersections
STEP 1
Rearrange both circle equations so that one side is zero
STEP 2
Put the circle equations equal to each other (i.e. solve simultaneously!)
STEP 3
Expand/rearrange/simplify into a linear equation
- The $x^{2}$ and $y^{2}$ terms will cancel, leaving an equation of the form $y = m x + c, x = k$ or $y = k$
  (These are 'diagonal line', 'vertical line' and 'horizontal line')
  The intersection(s) will lie on this line
STEP 4
Substitute the linear equation into either of the circle equations
Solving this equation will lead to either the $x$ -coordinate(s) or $y$ -coordinate(s) of the intersection(s)
STEP 5
Substitute the $x$ (or $y$ ) coordinates into either circle equation to find the corresponding $y$ (or $x$ ) coordinates
This step will not be needed in the case of the linear equation being of the form $x = k$ or $y = k$

Examiner Tip

Even if not given, or asked for, a sketch of the circles can help visualise their positions relative to each other
- You can then see if your final answers make sense with your sketch

Worked example

Determine the number of intersections between the circles with equations

x^{2} + y^{2} = 2

and

{(x - 4)}^{2} + y^{2} = 10

$x^{2} + y^{2} = 2$ has centre $(0, 0)$ and radius $\sqrt{2}$ .

${(x - 4)}^{2} + y^{2} = 10$ has centre $(4, 0)$ and radius $\sqrt{10}$ .

$\begin{array}{rcl} d^{2} & = & {(4 - 0)}^{2} + {(0 - 0)}^{2} \\ d & = & 4 \end{array}$

Using a sketch may help you to 'see' that $d = 4$ .

$r_{1} = \sqrt{2}, r_{2} = \sqrt{10}$

Compare $d$ with the sum and difference of $r_{1}$ and $r_{2}$ .

$\begin{array}{rcl} r_{1} + r_{2} & = & 4.576 . . . \\ r_{2} - r_{1} & = & 1.748 . . . \end{array}$

$\begin{array}{rcl} ∴ r_{2} - r_{1} & < & d < r_{1} + r_{2} \end{array}$

The circles intersect twice

Determine the coordinates of any intersections between the circles with equations

x^{2} + y^{2} = 2

and

{(x - 4)}^{2} + y^{2} = 10

STEP 1 - Rearrange both equations so zero is on one side

$\begin{array}{rcl} x^{2} + y^{2} - 2 & = & 0 \\ {(x - 4)}^{2} + y^{2} - 10 & = & 0 \end{array}$

STEP 2 - Put the equations equal to each other

$x^{2} + y^{2} - 2 = {(x - 4)}^{2} + y^{2} - 10$

STEP 3 - Expand and rearrange until in linear form

$\begin{array}{rcl} x^{2} + 8 & = & x^{2} - 8 x + 16 \\ 8 x - 8 & = & 0 \\ x & = & 1 \end{array}$

STEP 4 - Substitute into either circle equation

$\begin{array}{rcl} {(1)}^{2} + y^{2} & = & 2 \\ y^{2} & = & 1 \\ y & = & 1, y = - 1 \end{array}$

STEP 5 - Not required in this case

The intersections of the two circles have coordinates (1, 1) and (1,-1)

Equation of Common Chord

What is a common chord?

For circles that intersect twice the common chord is the line that joins the points of intersection
This line is a chord in both circles
- Circles that intersect once (touch) have a common tangent

Common chord of two circles goes between the intersections

How do I find the equation of a common chord?

As a common chord is a straight line, its equation will be of the form $y = m x + c$ unless
- it is a horizontal line, in which case its equation will be of the form $y = k$
- it is a vertical line, in which case its equation will be of the form $x = k$
Depending on the known information, there are two ways to find the equation of the common chord
- If the equations of the circles are known
  - Equate the equations and rearrange the equation into one of the three forms above
  - For example,
    $x^{2} + y^{2} - 4 x + 2 y - 8 = x^{2} + y^{2} - 8 x - 2 y + 16$
    $4 x + 4 y = 24$
    So the equation of the common chord is $y = 6 - x$
- If the points of intersection are known
  - Use the method of finding the equation of a straight line from two known points
  - If the intersection points are $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ then the equation of the common chord would be
    - $y - y_{1} = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} (x - x_{1})$
    - $\frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ is the gradient and it can be easier to work this out first, separately

Worked example

Two circles intersect at the points with coordinates $(3, - 1)$ and $(8, 4)$ .

Find the equation of the common chord of the two circles.

The points of intersection are known.
Use the method of finding the equation of a straight line from two known points.

First find the gradient,

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} = \frac{4 - (- 1)}{8 - 3} = \frac{5}{5} = 1$

Apply $y - y_{1} = m (x - x_{1})$ to get the equation of the common chord.

$\begin{array}{rcl} y - (- 1) & = & 1 (x - 3) \\ y + 1 & = & x - 3 \end{array}$

The equation of the common chord is $y = x - 4$

Intersection of Two Circles (Cambridge O Level Additional Maths)

Revision Note