1. Co-ordinate Geometry of the Circle Notes Aidan Roche 2009 Aidan Roche 2009 1(c) Aidan Roche 2009

2. Given the centre and radius of a circle, to find the equation of Circle K? K r Method Sub centre & radius into: (x – h) 2 + (y – k) 2 = r 2 If required expand to: x 2 + y 2 +2gx +2fy + c = 0 c(h, k) 2(c) Aidan Roche 2009

3. To find the centre and radius. Given the Circle K: (x – h) 2 + (y – k) 2 = r 2 Method Centre: c(h, k) Radius = r K r c 3(c) Aidan Roche 2009

4. To find the centre and radius. Given the Circle K: x 2 + y 2 = r 2 Method Centre: c(0, 0) Radius = r K r c 4(c) Aidan Roche 2009

5. To find centre and radius of K. G iven the circle K: x 2 + y 2 +2gx +2fy + c = 0 ? K Method Centre: c(-g, -f) Radius: r c 5(c) Aidan Roche 2009

6. Given equation of circle K, asked if a given point is on, inside or outside the circle? a Method Sub each point into the circle formula K = 0 Answer > 0 outside Answer = 0on Answer < 0inside b c K 6(c) Aidan Roche 2009

7. Important to remember Theorem Angle at centre is twice the angle on the circle standing the same arc c 7(c) Aidan Roche 2009 θ 2θ2θ a b d

8. Important to remember Theorem Angle on circle standing the diameter is 90 o diameter 8(c) Aidan Roche 2009 90 o

9. To find equation of circle K given end points of diameter? K Method Centre is midpoint [ab] Radius is ½|ab| Sub into circle formula a 9(c) Aidan Roche 2009 b c r

10. To prove a locus is a circle? Method If the locus of a set of points is a circle it can be written in the form: x 2 + y 2 +2gx + 2fy + c = 0 We then can write its centre and radius. c K 10 (c) Aidan Roche 2009 r

11. To find the Cartesian equation of a circle given Trigonometric Parametric equations? Method Trigonometric equations of a circle are always in the form: x = h ± rcosѲ y = k ± rsinѲ Sub h, k and r into Cartesian equation: (x – h) 2 + (y – k) 2 = r 2 c K 11 (c) Aidan Roche 2009 r

12. To prove that given Trigonometric Parametric equations (x = h ± rcosѲ, y = k ± rsinѲ) represent a circle? Method Rewrite cosѲ (in terms of x, h & r) and then evaluate cos 2 Ѳ. Rewrite sinѲ (in terms of y, h & r) and then evaluate sin 2 Ѳ. Sub into: sin 2 Ѳ + cos 2 Ѳ = 1 If it’s a circle this can be written in the form: x 2 + y 2 +2gx + 2fy + c = 0 c K 12 (c) Aidan Roche 2009 r

13. To find the Cartesian equation of circle (in the form: x 2 + y 2 = k) given algebraic parametric equations? Method Evaluate: x 2 + y 2 The answer = r 2 Centre = (0,0) & radius = r c K 13 (c) Aidan Roche 2009 r

14. Given equations of Circle K and Circle C, to show that they touch internally? K Method Find distance between centres If d = r 1 - r 2 QED C r1r1 r2r2 d 14(c) Aidan Roche 2009 c1c1 c2c2

15. Given equations of Circle K and Circle C, to show that they touch externally? K Method Find distance d between centres If d = r 1 + r 2 QED C r1r1 r2r2 d 15(c) Aidan Roche 2009 c1c1 c2c2

16. Given circle K and the line L to find points of intersection? a Method Solve simultaneous equations b L K 16(c) Aidan Roche 2009

17. Important to remember Theorem A line from the centre (c) to the point of tangency (t) is perpendicular to the tangent. c 17(c) Aidan Roche 2009 90 o Tangent K radius t

18. Important to remember Theorem A line from the centre perpendicular to a chord bisects the chord. c 18(c) Aidan Roche 2009 90 o a b radius d

19. Given equation of Circle K and equation of Tangent T, find the point of intersection? K T Method Solve the simultaneous equations 19(c) Aidan Roche 2009 t

20. Given equation of Circle K and asked to find equation of tangent T at given point t? K t Method 1 Find slope [ct] Find perpendicular slope of T Solve equation of the line c T Method 2 Use formula in log tables 20(c) Aidan Roche 2009

21. To find equation of circle K, given that x-axis is tangent to K, and centre c(-f, -g) ? X-axis Method On x-axis, y = 0 so t is (-f, 0) r = |f| Sub into circle formula c(-g, -f) K 21 (c) Aidan Roche 2009 t(-g, 0) r = |f|

22. To find equation of circle K, given that y-axis is tangent to K, and centre c(-f, -g) ? y-axis Method On y-axis, x = 0 so t is (0, -g) r = |g| Sub into circle formula c(-g, -f) K 22 (c) Aidan Roche 2009 t(0, -f) r = |g|

23. Given equation of Circle K and equation of line L, how do you prove that L is a tangent? K L Method 2 Find distance from c to L If d = r it is a tangent 23(c) Aidan Roche 2009 r Method 1 Solve simultaneous equations and find that there is only one solution c

24. Given equation of Circle K & Line L: ax + by + c = 0 to find equation of tangents parallel to L? K r Method 1 Find centre c and radius r Let parallel tangents be: ax + by + k = 0 Sub into distance from point to line formula and solve: c L 24(c) Aidan Roche 2009 T1T1 T2T2 r

25. Given equation of Circle K and point p, to find distance d from a to point of tangency? K c t Method Find r Find |cp| Use Pythagoras to find d p T r |cp| d? 25(c) Aidan Roche 2009

26. Given equation of Circle K and point p, to find equations of tangents from p(x 1,y 1 )? K c p T1T1 r 26(c) Aidan Roche 2009 T2T2 r Method 1 Find centre c and radius r Sub p into line formula and write in form T=0 giving: mx – y + (mx 1 – y 1 ) = 0 Use distance from point to line formula and solve for m:

27. Given equation of Circle K and Circle C, to find the common Tangent T? K T Method Equation of T is: K – C = 0 C 27(c) Aidan Roche 2009

28. Given equation of Circle K and Circle C, to find the common chord L? K L C Method Equation of T is: K – C = 0 28(c) Aidan Roche 2009

29. Given three points and asked to find the equation of the circle containing them? a Method Sub each point into formula: x 2 + y 2 + 2gx + 2fy + c = 0 Solve the 3 equations to find: g, f and c, Sub into circle formula b c 29(c) Aidan Roche 2009

30. Given 2 points on circle and the line L containing the centre, to find the equation of the circle? a Method Sub each point into the circle: x 2 + y 2 + 2gx + 2fy + c = 0 Sub (-g, -f) into equation of L Solve 3 equations to find: g, f and c, Sub solutions into circle equation b L 30(c) Aidan Roche 2009

31. Given the equation of a tangent, the point of tangency and one other point on the circle, to find the equation of the circle? a Method Sub each point into the circle: x 2 + y 2 + 2gx + 2fy + c = 0 Use the tangent & tangent point to find the line L containing the centre. Sub (-g, -f) into equation of L Solve 3 equations to find: g, f and c, Sub solutions into circle equation b T 31(c) Aidan Roche 2009 L