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 radius r 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 to x^2 + y^2 = r^2.

Determining the Centre and Radius

  • The centre (h, k) and radius r 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 right-hand 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.