MATHEMATICS WORKSHEET

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1 MATHEMATICS WORKSHEET XI Grade (Semester 1) Chapter 3 by: Ignatia Maria Midawati SMAK ST. ALBERTUS (ST. ALBERT Senior High School) Talang 1 Street Malang 6511, Indonesia Phone (0341) , Fax.(0341) Blog:

2 PREFACE Mathematics Module THE CIRCLE is written for students of St. Albert Senior High School at XI Grade Semester 1. The contents are arranged under some worksheets which the students can fill to learn about THE CIRCLE. Each worksheet is expanded in detail and step by step manner for easy understanding. It starts with a brief introduction and explanation follwed by filled examples and numerous simple exercises to build up the student s technical skills and to reinforce his or her understanding. It is hoped that this approach will enable the individual student working on his or her own to make effective use of the module besides enabling the teacher to use them with mixed ability groups. Finally, we would like to thank to all those involved in the production of this module. Ignatia Maria Midawati CONTENTS Page Term in Mathematics - The Circle... 3 Worksheet 1 The Equation of The Circle I... 4 Worksheet The Equation of The Circle II Worksheet 3 TANGENTS AND NORMALS Worksheet 4 POLES AND POLARS... 6 First published October, 008

3 BIBLIOGRAPHY 1. THE ELEMENTS OF ANALYTICAL GEOMETRY, J.T. Brown, M.A. B.Sc and C.W.M. Manson, M.A.. New Additional Mathematics, Ho Soo Thong, Dip Ed, Khor Nyak Hiong, Bsc, Dip Ed TERM in MATHEMATICS THE CIRCLE NO Mathematics How to read in English Expressions = 7 Five plus two is equal to seven. 9 3 = 6 Nine minus three is equal to six 3. 7 x 9 = 63 Seven times nine is equal to sixty three 4. 1 : 3 = 4 Twelve devided by three is equal to four x a half, a quarter, Two thirds, three fifths, two and one third,, Two over three, three over five, two and one over three 10 x + y x to the power of, x power Square root of five Square of five or five to the power two or five to the second power Square root of x to the power two plus y to the power two 3

4 Worksheet 1 st Topic : The Equation of The CIRCLE I TIME : 3 X 45 minutes STANDARD COMPETENCY : 3. To compose the equation of a circle and its tangent. BASIC COMPETENCY: 3.1 To compose the equation of a circle that is in compliance with the given conditions. In this chapter, you will learn about : How to determine the equation of the circle whose centre is the origin (0,0) with radius r. How to determine the equation of the circle whose centre is the point (h,k) with radius r. How to determine the centre and the radius of a known circle equation. A. The Equation of the circle whose centre is the origin (0,0) with radius r A circle is the locus of all points equidistant from a fixed point on the plane. The fixed point is called the centre of the circle and the equal distance is called the radius of the circle. Generally, the equation of the circle whose centre is origin with radius r is Y P(x,y) x + y = r X O x y Let P be any point (x,y) on the circle then with phytagoras theorem we get 4

5 So, we have x + y = r (because the length of OP is r for any P on the circle). This equation is satisfied at any point on the circle and at no other point and is therefore the equation of the circle. Example 1 Find the equation of the circle: a. with the origin as centre and of radius 3, b. with the origin as centre and of radius, c. with the origin as centre and of diameter 5, d. with the origin as centre and passing through the point (1,), e. with the points (-,0) and (,0) as ends of a diameter. a. the equation of the circle with the origin as centre and of radius 3, x + y = x + y = r KK b. with the origin as centre and of radius c. with the origin as centre and of diameter 5, d. with the origin as centre and passing through the point (1,), e. with the points (-,0) and (,0) as ends of a diameter. 5

6 Example Find the centre and length of radius of the circle: a. b. c. a. P ( K, K ), r = K b... P ( K, K ), r = K c... P ( K, K ), r = K B. The Equation of the circle whose centre is the point (h,k) with radius r Y P(x,y) The formula of a distance between points, A (x 1,y 1 ) and B (x,y ) is KK KK + KK K K AB = ( ) ( ) O A(h,k) X Let A be the centre point (a,b) and let P(x,y) be any point on the circle. Based on the formula above, we get the radius of the circle whose centre is the point ( h, k) is AP = r = ( x h) + ( y k) By squaring the equation above, we have ( ) ( ) x h + y k = r Because it is used by any points on the circle, the equation is applied for all points on the circle. So, generally the equation of the circle is formulated as ( x h) + ( y k) = r the centre point ( h, k) with radius = r Sometimes the equation of the circle is also formulated as 6

7 the centre point and the radius = r = Prove the formula! Example 3 Find the equation of the circle : a. with the point (,-3) as centre and of radius 3, b. with the point (5,-3) as centre and of diameter, c. with the point (-1,) as centre and passing through the point (-,0), 7

8 Example 4 Find the centre and length of radius of the circle: a. b. c. d. Example 5 A circle has centre at the point B(3,1) and radius 5. Find: a. the equation of the circle, b. the point of intersection of the circle and y axis, c. the point of intersection of the circle and x axis d. draw the sketch 8

9 Exercise 1 1. Find the equation of the circle: a. with the origin as centre and of radius, b. with the point (-3,1) as centre and of radius 5, c. with the point (-1,) as centre and of radius ½.. Find the equation of the circle: a. with the origin as centre and passing through the point (1,), b. with the point (1,) as centre and passing through the point (3,-1), c. with the points (-4,1), (,5) as ends of a diameter. 3. Find the centre and length of radius of the circle: a. ( 1) + ( y + 3) = 5 x x d. + ( y + ) = 49 b. ( x + ) + ( y + 4) = 36 e. ( x 4) + y = 1 c. ( x 3) + ( y + 3) = 9 4. Write the equation of the circle below into the formula ( ) ( ) the centre and length of radius of the circle: a. x + y x 6y + 6 = 0 b. x + y + 4x + y + 1 = 0 c. x + y + 4x 4y + 7 = 0 d. x + y + 8x y = Find the centre and length of radius of the circle: a. x + y + x + y = 0 b. x + y + 6y = 0 c. x + y + 3x 7 y 1 = 0 d. x + y 8x + 4y + 0 = 0 e. x + y + 1 = 0 f. 7x + 7y 108x 96y 40 = 0 5 x h + y k = r, then find 9