Coordinate geometry of the circle

Learning objectives

Write the equation of a circle with centre (a, b) and radius r as (x − a)² + (y − b)² = r², and convert the expanded form x² + y² + 2gx + 2fy + c = 0 to this form by completing the square to read off the centre and radius

Find the points where a straight line meets a circle by substituting the line into the circle equation, and use the discriminant of the resulting quadratic to decide whether the line cuts, touches or misses the circle

Find the equation of the tangent to a circle at a given point on it, using the fact that the tangent is perpendicular to the radius at that point

Find the length of a tangent from an external point, and the equation of a chord or of the perpendicular bisector of a chord, using the result that the perpendicular from the centre bisects the chord

Determine the relationship between two circles by comparing the distance between their centres with the sum and the difference of their radii, and find their points of intersection where they exist