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The Circle

A circle is defined as the locus of all points \(P(x,y)\) that are a constant distance from a given point.

ellipse

In the diagram below, for every position of P, CP = r.

circle diagram

From the distance formula:

distance formula circle equation

Equation of a Circle (Centre (a,b), Radius r)

circle centre diagram equation expanded equation
Example

Write the equation of the circle with centre (5, −6) and radius \(3\sqrt{3}\).

example 1
Example

Show that the point B(2,5) lies on the circle:

circle eqn working
Example

Find the equation of the circle on the line joining A(−2,5) to B(4,3) as diameter.

  • Plot the line AB and sketch the circle.
AB diagram
  • Find C, the midpoint of AB.
midpoint

The centre is C(1,4).

  • Find the radius using the distance formula.
radius
  • Substitute into the circle equation.
final equation

Expanded Form of a Circle

expanded form completing square centre radius final form
Example

Find the radius and centre of the circle:

eqn working 1 working 2
Example

Does the equation represent a circle?

eqn working
Example

Find a possible range for the value of k:

eqn working

Intersection of a Line and a Circle

Example

Find where the line \(y = 2x + 8\) meets the circle:

\(x^2 + y^2 + 4x + 2y - 20 = 0\)

Substitute \(y = 2x + 8\):

substitution solution 1 solution 2

The line and circle meet at (−6, −4) and (−2, 4).

intersection diagram

Tangents to Circles

Example

Show that the line \(3x + y = -10\) is a tangent to the circle:

\(x^2 + y^2 - 8x + 4y - 20 = 0\)

Steps:

  • Rearrange: \(y = -3x - 10\)
  • Substitute into the circle equation
substitution

Alternatively, use the discriminant:

disc 1 disc 2 tangent diagram