The Circle

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

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

   From the distance formula,

   

 

 

The equation of a circle centre C(a,b),with radius r.

 

 

Example

Write the equation of the circle with centre (5, -6) and radius 3√3.

                                         

Example

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

 

   

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 .

• Find C, the midpoint of AB.

Hence the centre of the circle is  C (1,4)

• Find the length of the radius, using the distance formula
  with the centre of the circle .

       

• Substitute into the circle equation

  

Expanded form of the equation of a circle

Example

Find the radius and centre of the circle with equation
 

 

Example

Does the equation

represent a circle ?

 

Example

  Find a possible range for the value of k in the circle

 

Intersection of a line and a circle

Example

Find where the line y=2x+8 meets the circle
  x² + y² + 4x + 2y -20 = 0

First,substitute y=2x+8 into the circle equation:

   

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

 

Tangents to Circles

 

Example

Show that the line 3x + y = -10 is a tangent to the circle
x² +y² -8x + 4y -20 = 0 and find the point of contact.

Steps

• Re-arrange formula

y = -3x -10

• Substitute into circle equation

Alternatively, use the discriminant at this point: