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

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Equation of a Circle

What is the equation of a circle?

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

 

Circle with centre (a,b) and radius r

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

    • Substitute the values into the formula

 

Finding the equation of a circle

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

  • Make sure it is in the form open parentheses x minus a close parentheses squared plus open parentheses y minus b close parentheses squared equals r squared

    • 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 and radius of a circle given its equation

Examiner Tips and Tricks

  • Remember that the numbers in the brackets have the opposite signs to the coordinates of the centre

    Eqn of Circle sign flip, A Level & AS Level Pure Maths Revision Notes

     

  • Don't forget to take the square root of the right-hand side of the equation when finding the radius

Worked Example

Eqn of Circle Example, A Level & AS Level Pure Maths Revision Notes

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Finding the Centre & Radius

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

  • The most useful equation of a circle is open parentheses x minus a close parentheses squared plus open parentheses y minus b close parentheses squared equals r squared

    • This is so the centre, open parentheses a comma space b close parentheses 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 squared plus y squared plus 2 g x plus 2 f y plus c equals 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 open parentheses a comma space b close parentheses and radius r

  • Rearranging to this form will often involve completing the square

Completing the square to find the centre and radius of a circle

Worked Example

Circle Ctr Rad Example, A Level & AS Level Pure Maths Revision Notes

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

The three cases for intersections between a circle and a lin

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

  • For the equation of a circle in the form open parentheses x minus a close parentheses squared plus open parentheses y minus b close parentheses squared equals r squared and the equation of a line in the form y equals m x plus c

  • STEP 1
    Substitute the linear equation into the circle equation

    • e.g. open parentheses x minus 5 close parentheses squared plus open parentheses y minus 2 close parentheses squared equals 13 and y equals x minus 4 would become open parentheses x minus 5 close parentheses squared plus open parentheses open parentheses x minus 4 close parentheses minus 2 close parentheses squared equals 13

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

    • e.g. open parentheses x minus 5 close parentheses squared plus open parentheses x minus 6 close parentheses squared equals 13 x squared minus 10 x plus 25 plus x squared minus 12 x plus 36 minus 13 equals 0 x squared minus 11 x plus 24 equals 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.   open parentheses x minus 3 close parentheses open parentheses x minus 8 close parentheses equals 0 x equals 3 comma space space x equals 8 Two solutions so the line and the circle intersect twice

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

    • e.g.

y equals x minus 4 y equals 3 minus 4 equals negative 1 comma space space y equals 8 minus 4 equals 4

The line and the circle intersect at the points open parentheses 3 comma space minus 1 close parentheses and open parentheses 8 comma space 4 close parentheses

Examiner Tips and Tricks

  • A horizontal or vertical line could intersect a circle, when the full method shown above is unnecessary

    • Horizontal lines have the form y equals k, so substitute k for y in the circle equation (and solve for x)

    • Vertical lines have the form x equals k, so substitute k for x in the circle equation (and solve for y)

Worked Example

Show that the line y equals 4 x plus 4 is tangent to the circle open parentheses x minus 4 close parentheses squared plus open parentheses y minus 3 close parentheses squared equals 17. State the coordinates of the point of intersection between the tangent and the circle.

STEP 1 - Substitute the linear equation into the circle equation

open parentheses x minus 4 close parentheses squared plus open parentheses 4 x plus 4 minus 3 close parentheses squared equals 17

STEP 2 - Rearrange to a quadratic

table row cell open parentheses x minus 4 close parentheses squared plus open parentheses 4 x plus 1 close parentheses squared minus 17 end cell equals 0 row cell x squared minus 8 x plus 16 plus 16 x squared plus 8 x plus 1 minus 17 end cell equals 0 row cell 17 x squared end cell equals 0 end table

STEP 3 - Solve

x squared equals 0

x equals 0 (repeated)

There is only one point of intersection so the line bold italic y bold equals bold 4 bold italic x bold plus bold 4 is tangent to the circle

STEP 4 - The coordinates are required

y equals 4 open parentheses 0 close parentheses plus 4 equals 4

The line and circle intersect at stretchy left parenthesis 0 comma space 4 stretchy right parenthesis

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