Revision NotesExam QuestionsPast Papers
IGCSEAdditional MathsCIERevision NotesCoordinate GeometryCoordinate Geometry of the CircleIntersection of Two Circles

Intersection of Two Circles (CIE IGCSE Additional Maths)

Revision Note

Paul

Author

Paul

Last updated

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 open parentheses x subscript 1 comma space y subscript 1 close parentheses and open parentheses x subscript 2 comma space y subscript 2 close parentheses, d squared equals open parentheses x subscript 2 minus x subscript 1 close parentheses squared plus open parentheses y subscript 2 minus y subscript 1 close parentheses squared
  • The radii of the two circles, r subscript 1 and r subscript 2, where r subscript 2 greater or equal than r subscript 1 are also needed
    • If r subscript 2 minus r subscript 1 less than d less than r subscript 1 plus r subscript 2 then the circles intersect twice
    •  If d equals r subscript 2 minus r subscript 1 or d equals r subscript 1 plus r subscript 2 then the circles intersect once
    • If d greater than r subscript 1 plus r subscript 2 or d less than r subscript 2 minus r subscript 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 squared and y squared terms will cancel, leaving an equation of the form y equals m x plus c comma space x equals k or y equals 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 equals k or y equals 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

a)
Determine the number of intersections between the circles with equations x squared plus y squared equals 2 and open parentheses x minus 4 close parentheses squared plus y squared equals 10.

x squared plus y squared equals 2 has centre open parentheses 0 comma space 0 close parenthesesand radius square root of 2.

open parentheses x minus 4 close parentheses squared plus y squared equals 10 has centre open parentheses 4 comma space 0 close parenthesesand radius square root of 10.

table attributes columnalign right center left columnspacing 0px end attributes row cell d squared end cell equals cell open parentheses 4 minus 0 close parentheses squared plus open parentheses 0 minus 0 close parentheses squared end cell row d equals 4 end table

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

r subscript 1 equals square root of 2 comma space r subscript 2 equals square root of 10

Compare d with the sum and difference of r subscript 1 and r subscript 2.

table row cell r subscript 1 plus r subscript 2 end cell equals cell 4.576 space... end cell row cell r subscript 2 minus r subscript 1 end cell equals cell 1.748 space... end cell end table

table row cell therefore space r subscript 2 minus r subscript 1 end cell less than cell d less than r subscript 1 plus r subscript 2 end cell end table

The circles intersect twice

b)
Determine the coordinates of any intersections between the circles with equations x squared plus y squared equals 2 and open parentheses x minus 4 close parentheses squared plus y squared equals 10.

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

table row cell x squared plus y squared minus 2 end cell equals 0 row cell open parentheses x minus 4 close parentheses squared plus y squared minus 10 end cell equals 0 end table

STEP 2 - Put the equations equal to each other

x squared plus y squared minus 2 equals open parentheses x minus 4 close parentheses squared plus y squared minus 10

STEP 3 - Expand and rearrange until in linear form

table row cell x squared plus 8 end cell equals cell x squared minus 8 x plus 16 end cell row cell 8 x minus 8 end cell equals 0 row x equals 1 end table

STEP 4 - Substitute into either circle equation

table row cell open parentheses 1 close parentheses squared plus y squared end cell equals 2 row cell y squared end cell equals 1 row y equals cell 1 comma space y equals negative 1 end cell end table

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 equals m x plus c unless
    • it is a horizontal line, in which case its equation will be of the form y equals k
    • it is a vertical line, in which case its equation will be of the form x equals 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 squared plus y squared minus 4 x plus 2 y minus 8 equals x squared plus y squared minus 8 x minus 2 y plus 16
        4 x plus 4 y equals 24
        So the equation of the common chord is y equals 6 minus 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 open parentheses x subscript 1 comma space y subscript 1 close parentheses and open parentheses x subscript 2 comma space y subscript 2 close parentheses then the equation of the common chord would be
        • y minus y subscript 1 equals fraction numerator y subscript 2 minus y subscript 1 over denominator x subscript 2 minus x subscript 1 end fraction open parentheses x minus x subscript 1 close parentheses
        • fraction numerator y subscript 2 minus y subscript 1 over denominator x subscript 2 minus x subscript 1 end fraction is the gradient and it can be easier to work this out first, separately

Worked example

Two circles intersect at the points with coordinates open parentheses 3 comma space minus 1 close parentheses and open parentheses 8 comma space 4 close parentheses.

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 equals fraction numerator y subscript 2 minus y subscript 1 over denominator x subscript 2 minus x subscript 1 end fraction equals fraction numerator 4 minus open parentheses negative 1 close parentheses over denominator 8 minus 3 end fraction equals 5 over 5 equals 1

Apply y minus y subscript 1 equals m open parentheses x minus x subscript 1 close parentheses to get the equation of the common chord.

table attributes columnalign right center left columnspacing 0px end attributes row cell y minus open parentheses negative 1 close parentheses end cell equals cell 1 open parentheses x minus 3 close parentheses end cell row cell y plus 1 end cell equals cell x minus 3 end cell end table

The equation of the common chord is bold italic y bold equals bold italic x bold minus bold 4

You've read 0 of your 5 free revision notes this week

Sign up now. It’s free!

Join the 100,000+ Students that ❤️ Save My Exams

the (exam) results speak for themselves:

    Test yourself

    Did this page help you?

    Previous:Tangents to CirclesNext:Radian Measure
    Paul

    Author: Paul

    Expertise: Maths

    Paul has taught mathematics for 20 years and has been an examiner for Edexcel for over a decade. GCSE, A level, pure, mechanics, statistics, discrete – if it’s in a Maths exam, Paul will know about it. Paul is a passionate fan of clear and colourful notes with fascinating diagrams – one of the many reasons he is excited to be a member of the SME team.