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www.maths4scotland.co.uk Higher Maths Strategies The Circle Click to start Maths4Scotland Higher The following questions are on The Circle Non-calculator questions will be indicated You will need a pencil, paper, ruler and rubber. Click to continue Maths4Scotland Higher Find the equation of the circle with centre (–3, 4) and passing through the origin. Find radius (distance formula): You know the centre: Write down equation: r 5 (3, 4) ( x 3)2 ( y 4)2 25 Hint Previous Quit Quit Next Maths4Scotland Higher Explain why the equation x2 does not represent a circle. Consider the 2 conditions y2 2x 3 y 5 0 1. Coefficients of x2 and y2 must be the same. 2. Radius must be > 0 g 1, Calculate g and f: Evaluate Deduction: g f c 2 2 f g 2 f 2 c 0 so 3 2 (1) 2 i.e. g 2 f 2 c 0 3 2 2 5 1 4 1 2 5 0 g 2 f 2 c not real Equation does not represent a circle Hint Previous Quit Quit Next Maths4Scotland Higher Find the equation of the circle which has P(–2, –1) and Q(4, 5) as the end points of a diameter. Q(4, 5) C Make a sketch P(-2, -1) (1, 2) Calculate mid-point for centre: Calculate radius CQ: Write down equation; r 18 x 1 y 2 18 2 2 Hint Previous Quit Quit Next Maths4Scotland Higher Find the equation of the tangent at the point (3, 4) on the circle x2 y 2 2x 4 y 15 0 Calculate centre of circle: P(3, 4) (1, 2) Make a sketch O(-1, 2) Calculate gradient of OP (radius to tangent) Gradient of tangent: m 2 Equation of tangent: y 2 x 10 m 1 2 Hint Previous Quit Quit Next Maths4Scotland Higher The point P(2, 3) lies on the circle ( x 1)2 ( y 1)2 13 Find the equation of the tangent at P. Find centre of circle: P(2, 3) (1, 1) Make a sketch O(-1, 1) Calculate gradient of radius to tangent Gradient of tangent: 3 m 2 Equation of tangent: 2 y 3x 12 m 2 3 Hint Previous Quit Quit Next Maths4Scotland Higher O, A and B are the centres of the three circles shown in the diagram. The two outer circles are congruent, each touches the smallest circle. Circle centre A has equation x 12 2 y 5 25 2 The three centres lie on a parabola whose axis of symmetry is shown the by broken line through A. a) i) State coordinates of A and find length of line OA. ii) Hence find the equation of the circle with centre B. b) The equation of the parabola can be written in the form y px( x q) A is centre of small circle A(12, 5) Find OA (Distance formula) Use symmetry, find B B(24, 0) Find radius of circle A from eqn. Find radius of circle B 13 5 8 Eqn. of B Points O, A, B lie on parabola – subst. A and B in turn 0 24 p(24 q) 5 12 p(12 q) Solve: Find p and q. 13 5 ( x 24)2 y 2 64 p 5 , 144 q 24 Hint Previous Quit Quit Next Maths4Scotland Higher Circle P has equation x2 y 2 8x 10 y 9 0 Circle Q has centre (–2, –1) and radius 22. a) i) Show that the radius of circle P is 42 ii) Hence show that circles P and Q touch. b) Find the equation of the tangent to circle Q at the point (–4, 1) c) The tangent in (b) intersects circle P in two points. Find the x co-ordinates of the points of intersection, expressing your answers in the form a b 3 Find centre of circle P: Find radius of circle :P: (4, 5) Find distance between centres 72 6 2 Gradient of radius of Q to tangent: Equation of tangent: m 1 Deduction: = sum of radii, so circles touch Gradient tangent at Q: m 1 y x5 2 2 Solve eqns. simultaneously x y 8 x 10 y 9 0 y x5 Previous 42 52 9 32 4 2 Quit Quit Soln: 22 3 Hint Next Maths4Scotland Higher For what range of values of k does the equation represent a circle ? g 2k , Determine g, f and c: State condition g f c 0 2 2 5k 2 k 2 0 Simplify f k, x2 y 2 4kx 2ky k 2 0 c k 2 Put in values (2k )2 k 2 (k 2) 0 Need to see the position of the parabola Complete the square 1 5 5 k2 k 2 5k 1 5 k 10 1 10 2 2 Previous 1 2 100 195 100 Minimum value is 195 1 when k 100 10 This is positive, so graph is: Expression is positive for all k: So equation is a circle for all values of k. Quit Quit Hint Next Maths4Scotland Higher For what range of values of c does the equation represent a circle ? Determine g, f and c: g 3, State condition g2 f 2 c 0 Simplify 94c 0 Re-arrange: x2 y 2 6 x 4 y c 0 f 2, c? Put in values 32 (2)2 c 0 c 13 Hint Previous Quit Quit Next Maths4Scotland Higher The circle shown has equation ( x 3)2 ( y 2)2 25 Find the equation of the tangent at the point (6, 2). Calculate centre of circle: (3, 2) Calculate gradient of radius (to tangent) 4 m 3 3 4 Gradient of tangent: m Equation of tangent: 4 y 3x 26 Hint Previous Quit Quit Next Maths4Scotland Higher When newspapers were printed by lithograph, the newsprint had to run over three rollers, illustrated in the diagram by 3 circles. The centres A, B and C of the three circles are collinear. The equations of the circumferences of the outer circles are ( x 12)2 ( y 15)2 25 and ( x 24)2 ( y 12)2 100 Find the equation of the central circle. Find centre and radius of Circle A (12, 15) Find centre and radius of Circle C (24, 12) Find diameter of circle B 45 (5 10) 30 Use proportion to find B 25 27 15, 45 Previous (4, 3) r 5 25 r 10 Equation of B Quit 27 B 20 362 272 45 Find distance AB (distance formula) Centre of B (24, 12) (-12, -15) 36 so radius of B = 15 25 36 20 relative to C 45 x 4 y 3 225 2 Quit 2 Next Hint Maths4Scotland Higher You have completed all 11 questions in this presentation Previous Quit Quit Back to start Maths4Scotland Higher Table of exact values sin cos tan Return 30° 45° 60° 6 1 2 4 3 1 2 1 2 3 2 1 2 3 2 1 3 1 3