Coordinate Geometry of the Circle

You will be able to

• Write the equation of a circle with centre (a, b) and radius r as (x − a)² + (y − b)² = r², and expand it to the form x² + y² + 2gx + 2fy + c = 0.
• Find the centre and radius of a circle from its expanded equation by completing the square for x and for y.
• Determine whether a point lies inside, on or outside a circle by comparing (x − a)² + (y − b)² with r².

You will be able to

• Find where a line meets a circle by substituting y = mx + k into the circle's equation and solving the quadratic in x, giving the points exactly.
• Use the discriminant of that quadratic to decide whether the line is a chord (b² − 4ac > 0), a tangent (= 0) or misses the circle (< 0).
• Find the equation of the tangent at a point on a circle, using the fact that the tangent is perpendicular to the radius there.

You will be able to

• Calculate the distance d between the centres of two circles exactly, and compare it with r₁ + r₂ and |r₁ − r₂|.
• Deduce whether two circles are separate, touch externally, intersect at two points, touch internally or lie one inside the other.
• Find the equation of the common chord of two intersecting circles by subtracting their equations, and use it to find the points of intersection.