Duration: 9 hours Subtopic: 1.1 Introduction to Conic Sections 1.2 Circles 1.3 Parabolas 1.4 Ellipses 1.5 Hyperbolas 1.6 The Intersection of Straight Line and Conic Sections 1.7 Parametric Representations of Conic Sections

(a) Understand the meaning of conic sections (b) Define a circle (c) Determine the centre and radius of a circle by completing the square LEARNING OBJECTIVES

A conic section is a curve formed by the intersection of a plane and cone. By changing the angle of intersection, we can produce a: (i) Circle (ii) Parabola (iii) Ellipse (iv) Hyperbola

CIRCLE

PARABOLA

ELLIPSE

HYPERBOLA

A circle is a set of all points in a plane equidistant from a given fixed point called the centre. A line segment determined by joining the centre and any point on the circle is called a radius. Circle r C(h,k) y x k h

Let r represents the radius C(h,k) represents the centre r C(h,k) P(x,y) Standard equation of a circle x y Q Pythagoras Theorem (CQ) 2 + (PQ) 2 =(PC) 2

The equation of the circle with centre (0,0) and radius r is

Example 1: Determine the centre, radius and draw the circle represented by the following equation;

Solution : So, centre is at (4,-3) and the radius is C(4,-3) x y 0

Example 2 : Determine the equation of a circle having its centre at (3,-1) and passing through the point (-1,2). Solution : Radius, So the equation is

General equation of a circle Where, the centre ( h,k ) = (- g,- f ) From standard form:

General form: Where, and

Example 3 : Find the centre and radius of a circle given the equation of circle is

By completing the square; Simplify the result, Centre: (1,3) Radius = 2 Solution:

Comparing with ; ( h,k ) = ( - g,- f ) = (1,3) and 2g = - 2, g = -1 2f = - 6, f = g = 1, - f = 3 Centre: (1,3) Radius = 2 Alternative Method :

General equation of a circle Standard equation of a circle