C2: Coordinate Geometry of the Circle Learning Objective: To be able to find and use the equation of a circle.

Starter: Copy and complete: 1.The angle in a semi circle is ……….. 2.The perpendicular from the centre to a chord ……………. the chord. 3.The radius meets the ……………. at 90 0.

A circle is the locus of a point which moves in a plane so that its distance from a fixed point in the plane is constant. The Circle The fixed point is called the CENTRE and the constant distance is the RADIUS. A circle with centre (0, 0) and radius r has equation: x 2 + y 2 = r 2

The most general equation of a circle is (x – a) 2 + (y – b) 2 = r 2 This is equivalent to x 2 + y 2 – 2ax – 2by + a 2 + b 2 – r 2 = 0 ● (a,b) (x,y) r

Example 1 : Write down the equation of the circle with centre (5, 7) and radius 4. (x - a) 2 + (y -b) 2 = r 2 So the equation required is: (x - 5) 2 + (y -7) 2 = 4 2

Task 1: Exercise 4D, q 1&2

Example 2 Find the centre and radius of the circle x 2 + y 2 − 10x + 12y − 3 = 0 Example 3 Find the centre and radius of the circle 2x 2 + 2y x − 5y + 20 = 0

Task 2: Exercise 4D, q 3+

Example 3 The points A (3, 4) and B (9, 12) are at the ends of a diameter of a circle. Find the equation of the circle.

Task 3: Exercises 4E and 4F