Co-ordinate Geometry for Edexcel C2

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Presentation transcript:

Co-ordinate Geometry for Edexcel C2

What you need to know How to use the radius and centre of a circle to find its equation How to use the equation to find radius and centre of a circle To use angle properties of circles to solve problems in co-ordinate geometry Angle in a semicircle is 90⁰ Angle between radius and tangent is 90⁰ Perpendicular from the centre to a chord bisects the chord

Toolkit Distance formula Gradient formula Circle properties Radius at 90⁰ to tangent Angle in semi-circle is 90⁰ Midpoint formula Equation of a line Equation of a circle centre (a, b) radius r Solve the equation Simultaneous equations Quadratic formula Parallel Same gradient Crosses x - axis when y = 0 y - axis when x = 0 Complete the square Perpendicular Factorise Other end of a diameter Use vectors

Example 1 The points A and B have coordinates (–2, 11) and (8, 1) respectively. Given that AB is a diameter of the circle C, show that the centre of C has coordinates (3, 6) C is the midpoint

The points A and B have coordinates (–2, 11) and (8, 1) respectively The points A and B have coordinates (–2, 11) and (8, 1) respectively. (b) find an equation for C. Diameter Radius Equation

(c) Verify that the point (10, 7) lies on C (c) Verify that the point (10, 7) lies on C. Find the distance of P from the centre Distance = radius, so point P lies on the circle

(d) Find an equation of the tangent to C at the point (10, 7), giving your answer in the form y = mx + c, where m and c are constants. Tangent perpendicular to the radius Gradient of the radius Gradient of the tangent Equation of the line

Example 2 (part question) The diagram shows a sketch of the circle C with centre N and equation (a) Write down the coordinates of N. (b) Find the radius of C. N is the centre. Compare with

Example 2 The diagram shows a sketch of the circle C with centre N and equation The chord AB of C is parallel to the x-axis, lies below the x-axis and is of length 12 (c) Find the coordinates of A and B. y co-ords x co-ords

Summary How to use the radius and centre of a circle to find its equation How to use the equation to find radius and centre of a circle To use angle properties of circles to solve problems in co-ordinate geometry Angle in a semicircle is 90⁰ Angle between radius and tangent is 90⁰ Perpendicular from the centre to a chord bisects the chord