Equation of a Circle (Cambridge International IGCSE Further Maths)

Equation of a circle

What is the equation of a circle?

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

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

• 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 $( x - a )^{2} + ( y - b )^{2} = r^{2}$

  • 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

Exam Hint

• Remember that the numbers in the brackets have the opposite signs to the coordinates of the centre
• Don't forget to take the square root of the right-hand side of the equation when finding the radius

Worked Example

Finding the Centre & Radius

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

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

  • This is so the centre, $( a , b )$ 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 $( a , b )$ and radius $r$

• Rearranging to this form will often involve completing the square

Worked Example

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

• STEP 1 Substitute the linear equation into the circle equation

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

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

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

• 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. $( x - 3 ) ( x - 8 ) = 0 x = 3 , x = 8$ Two solutions so the line and the circle intersect twice

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

  • e.g.

$y=x-4\,y=3-4=-1,y=8-4=4$

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

Exam Hint

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

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

Worked Example Show that the line $y=4x+4$ is tangent to the circle $( x - 4 )^{2} + ( y - 3 )^{2} = 17$. State the coordinates of the point of intersection between the tangent and the circle.

STEP 1 - Substitute the linear equation into the circle equation

$( x - 4 )^{2} + ( 4 x + 4 - 3 )^{2} = 17$

STEP 2 - Rearrange to a quadratic

$\begin{gathered} ( x - 4 )^{2} + ( 4 x + 1 )^{2} - 17 & = 0 \\ x^{2} - 8 x + 16 + 16 x^{2} + 8 x + 1 - 17 & = 0 \\ 17 x^{2} & = 0 \end{gathered}$

STEP 3 - Solve

$x^{2}=0$

$x=0$ (repeated)

Final answer: There is only one point of intersection so the line$y=4x+4$is tangent to the circle $(x-4)^{2}+(y-3)^{2}=17$

STEP 4 - The coordinates are required

$y = 4 ( 0 ) + 4 = 4$

Final answer: The line and circle intersect at $(0,4)$