Intersection of Two Circles (Cambridge (CIE) O Level Additional Maths): Revision Note

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 $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$, $d^2={\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2$

• The radii of the two circles, $r_1$ and $r_2$, where $r_2\ge r_1$ 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

• 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 $y=mx+c,x=k$ 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 $y=k$

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

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 $y=mx+c$ 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, $x^2+y^2-4x+2y-8=x^2+y^2-8x-2y+16$ $4x+4y=24$ 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 $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ then the equation of the common chord would be

      • $y-y_1=\frac{y_2-y_1}{x_2-x_1}\left(x-x_1\right)$

      • $\frac{y_2-y_1}{x_2-x_1}$ is the gradient and it can be easier to work this out first, separately