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Co-ordinate Geometry of the Circle Notes

Co-ordinate Geometry of the Circle Notes. Aidan Roche 2009. Given the centre and radius of a circle, to find the equation of Circle K?. K. 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. r. c(h, k).

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Co-ordinate Geometry of the Circle Notes

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

  2. Given the centre and radius of a circle, to find the equation of Circle K? K • Method • Sub centre & radius into: (x – h)2 + (y – k)2 = r2 • If required expand to: x2 + y2 +2gx +2fy + c = 0 • r • c(h, k) (c) Aidan Roche 2009

  3. To find the centre and radius. Given the Circle K:(x – h)2 + (y – k)2 = r2 K • r • Method • Centre: c(h, k) • Radius = r • c (c) Aidan Roche 2009

  4. To find the centre and radius. Given the Circle K:x2 + y 2 = r2 K • r • Method • Centre: c(0, 0) • Radius = r • c (c) Aidan Roche 2009

  5. To find centre and radius of K. Given the circle K: x2 + y2 +2gx +2fy + c = 0? K • r • Method • Centre: c(-g, -f) • Radius: • c (c) Aidan Roche 2009

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

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

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

  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 • r • b • a • c (c) Aidan Roche 2009

  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: • x2 + y2 +2gx + 2fy + c = 0 • We then can write its centre and radius. • r • c • K (c) Aidan Roche 2009

  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 = r2 • r • c • K (c) Aidan Roche 2009

  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 cos2Ѳ. • Rewrite sinѲ (in terms of y, h & r) and then evaluate sin2Ѳ. • Sub into: sin2Ѳ + cos2Ѳ = 1 • If it’s a circle this can be written in the form: • x2 + y2 +2gx + 2fy + c = 0 • r • c • K (c) Aidan Roche 2009

  13. To find the Cartesian equation of circle (in the form: x2 + y2 = k) given algebraic parametric equations? • Method • Evaluate: x2 + y2 • The answer = r2 • Centre = (0,0) & radius = r • r • c • K (c) Aidan Roche 2009

  14. Given equations of Circle K and Circle C, to show that they touch internally? • C • Method • Find distance between centres • If d = r1 - r2QED • r2 • r1 K • c2 • d • c1 (c) Aidan Roche 2009

  15. Given equations of Circle K and Circle C, to show that they touch externally? • C • c2 • Method • Find distance d between centres • If d = r1 + r2QED • r2 K • r1 • d • c1 (c) Aidan Roche 2009

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

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

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

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

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

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

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

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

  24. Given equation of Circle K & Line L: ax + by + c = 0 to find equation of tangents parallel to L? • L • 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: • T1 • r • c K • r • T2 (c) Aidan Roche 2009

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

  26. Given equation of Circle K and point p, to find equations of tangents from p(x1,y1)? • T1 • Method 1 • Find centre c and radius r • Sub p into line formula and write in form T=0 giving: • mx – y + (mx1 – y1) = 0 • Use distance from point to line formula and solve for m: • p • r K • c • r • T2 (c) Aidan Roche 2009

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

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

  29. Given three points and asked to find the equation of the circle containing them? • b • Method • Sub each point into formula: • x2 + y2 + 2gx + 2fy + c = 0 • Solve the 3 equations to find: g, f and c, • Sub into circle formula • a • c (c) Aidan Roche 2009

  30. Given 2 points on circle and the line L containing the centre, to find the equation of the circle? • b • a • Method • Sub each point into the circle: • x2 + y2 + 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 L (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? • L • a • Method • Sub each point into the circle: • x2 + y2 + 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 (c) Aidan Roche 2009

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