C: Coordinate geometry in the (x, y) plane: Understand and use the coordinate geometry of the circle including using the equation of a circle in the form (x − a)² + (y − b)² = r²; completing the square to find the centre and radius of a circle; use of the following properties: • the angle in a semicircle is a right angle • the perpendicular from the centre to a chord bisects the chord • the radius of a circle at a given point on its circumference is perpendicular to the tangent to the circle at that point.

The idea

This topic covers circles in the coordinate plane. The standard equation (x − a)² + (y − b)² = r² gives centre (a, b) and radius r. If the equation is expanded, complete the square to rewrite it in standard form. For example, x² + y² − 6x + 8y − 11 = 0 becomes (x − 3)² + (y + 4)² = 36, so centre (3, −4), radius 6. You must also use three geometric properties: the angle in a semicircle is a right angle; the perpendicular from the centre to a chord bisects the chord; and the radius at a point is perpendicular to the tangent there. These allow you to find centres, radii, tangents and intersections. You need to apply these ideas to solve problems, often combining algebra and geometry.

What earns the marks

• Write down the centre and radius directly from an equation in the form (x − a)² + (y − b)² = r².
• Complete the square for both x and y terms to convert a general circle equation into standard form.
• Use the property that the angle in a semicircle is a right angle to deduce perpendicularity or to set up equations.
• Apply the fact that the perpendicular from the centre to a chord bisects the chord to find midpoints or lengths.
• Use the radius–tangent perpendicularity to find the equation of a tangent or to locate a centre.
• Solve problems involving intersections of lines and circles, using substitution and the discriminant where appropriate.

Common mistakes and how to put them right

• Forgetting to change the sign when reading the centre from (x − a)² + (y − b)² = r².

  Put it right: Compare with the standard form and note that the signs are opposite to those in the brackets.

• Completing the square incorrectly, such as forgetting to add the same constant to both sides or mishandling the coefficient of x².

  Put it right: Ensure the coefficient of x² and y² is 1 before completing the square; if not, divide through first.

• Assuming that any chord is bisected by the centre or that any line from the centre is perpendicular to a chord.

  Put it right: Only the perpendicular from the centre to a chord bisects it; conversely, the line from the centre to the midpoint of a chord is perpendicular to it.

Exam tips

• When given a circle equation, always rewrite it in standard form by completing the square; this makes the centre and radius obvious.
• For tangent problems, remember that the radius to the point of tangency is perpendicular to the tangent, so the product of their gradients is −1.
• Use the angle in a semicircle property to identify right angles in circle geometry problems, which can simplify finding lengths or equations.
• Check whether the circle equation is given in expanded form; if so, completing the square is usually required before answering.