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 .
The equation is
Example
Show that the point B (2,5) lies on the circle
Example
1. Plot the line AB and sketch the circle .
2. Find C, the midpoint of AB.
3. Find the length of the radius, using the distance formula with the centre of the circle .
4. Substitute into the circle equation
· Expanded form of the equation of a circle
Examples
Q1. Find the radius and centre of the circle
with equation
Q2. Does the equation
represent a circle ?
Q3. Find a possible range for the value of k
in the circle
· Intersection
of a line and a circle
To find where the line y=2x+8 meets the
circle
Substitute y=2x+8 into the circle
· Tangents to Circles
Example
Show that the line 3x + y =
-10 is a tangent to the circle
