1 OBJECTIVES : 4.1 CIRCLES (a) Determine the equation of a circle. (b) Determine the centre and radius of a circle by completing the square (c) Find the points of intersection of two circles (d) Find the equation of tangents and normal to a circle (e) Find the length of a tangent from a point to a circle 4.0 CONIC SECTIONS

2 A circle is a set of all points in a plane equidistant from a given fixed point called the center. The distance from the center to any point on the circle is called a radius. Definition

3 The Equation of a Circle in Standard Form 1) Center (0,0), radius : r Equation of a circle : (0,0) P(x, y) r Proof: x y

If a center of the circle is not located at the origin but at any arbitrary point ( h,k ), the equation becomes : x y (h,k) 4

5 2) Center ( h, k ), radius ; r Equation of a circle : C (h, k) r y x Proof: P (x,y)

From the Standard Equation, Expand and rearrange equation (1), Then substitute ; We get General Equation, The Equation of a Circle in General Form 6

From the General Equation By completing the square; Centre, Radius, By comparing to Standard Equation 7

8 Find the general equation of the circle with centre (- 2,3 ) and radius 5. Example 1 Solution

Determine the center and radius of a circle by completing the square. Example 2 Solution

Find the center of the circle and the length of it’s radius 10 Example 3 Solution compare with general equation :

Find the general equation of the circle with center (0,7) and touches the line 3x = y Example 4 Solution C (0,7) r Shortest distance = d

Given that a circle passes through (9, -7), (-3, -1) and (6,2 ). Find its equation. 13 Example 5 Solution General equation of circle

Find the equation of circle passing through the points (1,1), (3,2) and with the equation of diameter y-3x+7 = Example 6 Solution (1,1) (3,2) General equation : (1) (2)

y–3x+7=0 C(-g,-f) (1,1) (3,2) The diameter must passes through the centre, (-g,-f ) (3)

17 The points A and B have coordinates ( x 1, y 1 ) and (x 2, y 2 ). Show that the equation of a circle where AB is the diameter of a circle is Hence, find the equation of the circle where AB is a diameter and the points of A and B are ( 1, 1 ) and ( 2, 3). Example 7

18 A (x 1,y 1 ) B (x 2, y 2 ) P (x, y) AP and PB is perpendicular : Solution

19 A (x 1,y 1 ) B (x 2,y 2 ) P (x,y)

21 Important Notes Given the general equation of two circles: If Two circles intersect at two distinct points

22 Two circles touch to each other Two circles do not intersect to each other

Find the intersection points between the two circles below: 23 Example 8

Solution

25 Example 9

26 Solution

28 (1) Standard equation : x 2 + y 2 = r 2 The equation of tangents to a circle at the point P ( x 1, y 1 ) is given by xx 1 + yy 1 = r 2 0 P(x 1, y 1 ) x y The equations of tangents and normal to a circle xx 1 + yy 1 = r 2

29 (2) General equation : xx 1 + yy 1 + g (x + x 1 ) + f (y + y 1 ) +c = 0 The equation of tangents to a circle at the point P ( x 1, y 1 ) is given by

Find tangent and normal line to the circle x 2 + y 2 = 5 at the point ( -2, 1). 30 Example 10 Solution The equation of tangent at ( -2,1) xx 1 + yy 1 = r 2 -2x + y(1) = 5 y = 2x + 5

31 The equation of normal at ( -2,1)

32 Find the equation of the tangent and normal of the circle at the point (3,1) Example 11 Solution

Equation of tangent at (3,1), Equation of normal at (3,1),

34 Theorem The length of the tangent from a fixed point P(x 1, y 1 ) to a circle with equation x 2 + y 2 +2gx + 2fy+ c = 0 is given by C r P (x 1,y 1 ) d Q The length of a tangent from a point to a circle

Find the length of the tangent from the point A (6,7) to a circle x 2 + y 2 – 2x – 8y = 8 Example 12 Solution

Find the length of a tangent from the point P(5,3) to a circle 2x 2 + 2y 2 – 7x + 4y = 0 Example 13 Solution