The Coordinate Geometry of Circles
The Coordinate Geometry of Circles
Understanding Circles in Coordinate Geometry

In coordinate geometry, a circle is defined as the locus of all points
(x, y)
that are equidistant from a fixed point(h, k)
, known as the centre of the circle. 
The fixed distance from the centre of the circle to any point on the circle is called the radius of the circle.
Circle Equation

The standard form of the equation of a circle with centre
(h, k)
and radiusr
is(x  h)^2 + (y  k)^2 = r^2
. 
In the special case where the circle’s centre is at the origin
(0, 0)
, the equation simplifies tox^2 + y^2 = r^2
.
Determining the Centre and Radius

The centre
(h, k)
and radiusr
of a circle can be found directly from the equation of the circle. 
If the equation of the circle is given in the standard form, reading off the centre and the radius is straightforward. The centre is given by (h, k) and the radius r is the square root of the constant term on the righthand side.
Tangents to Circles

A line tangent to a circle is a straight line that just touches the circle, intersecting it at exactly one point called the point of tangency.

If a line is tangent to a circle at a point
(x1, y1)
, then the radius to the point of tangency is perpendicular to the tangent line.
Remember to visualise and plot these concepts on a graph as it will provide a more intuitive understanding of the topics covered.