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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

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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

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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

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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

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

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

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

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

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

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

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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

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

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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)

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

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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

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

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

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

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

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

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Solution

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25 Example 9

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26 Solution

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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

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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

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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

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

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

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

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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

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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

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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

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