Circle Final Strategies

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Circle Final Strategies

  1. 1. Exam Strategies Circles
  2. 2. What do you need to know? <ul><li>How to find centre and radius of circle </li></ul><ul><li>Finding equation of circle : (x-a) 2 + (y-b) 2 = r 2 </li></ul><ul><li>Equation of tangent to circle at given point </li></ul><ul><li>Proving tangency </li></ul><ul><li>Do circles touch? </li></ul><ul><li>Points of intersection (line & circle) </li></ul><ul><li>Common tangent </li></ul><ul><li>Midpoint of diameter is centre of circle </li></ul>
  3. 3. <ul><li>1. Finding centre and equation of circle </li></ul>Time remaining 00 :00 Time remaining 00 :05 Time remaining 00 :10 Time remaining 00 :20 Time remaining 00 :30 Time remaining 01 :00 Time remaining 02 :00 Centre is (2, 2) Radius is 2 2 4 2 2 2 Centre B is (10, 6) Radius is 2 (identical to A) Time remaining 03 :00 Time remaining 04 :00
  4. 4. <ul><li>2. Radius of circle </li></ul>Time remaining 00 :00 Time remaining 00 :05 Time remaining 00 :10 Time remaining 00 :20 Time remaining 00 :30 Time remaining 01 :00 Time remaining 02 :00 Time remaining 03 :00 For circle radius must be positive so: 2g = -6 2f = 4 g = -3 f = 2
  5. 5. <ul><li>3. Equation of tangent </li></ul>Time remaining 00 :00 Time remaining 00 :05 Time remaining 00 :10 Time remaining 00 :20 Time remaining 00 :30 Time remaining 01 :00 Time remaining 02 :00 Time remaining 03 :00 Time remaining 04 :00 Equation thru (3,4) Gradient of radius: Gradient of tangent:
  6. 6. <ul><li>4. Proving tangency / Common tangents </li></ul>8 Marks <ul><li>Show that the line x + 2y = 3 is a tangent to the circle </li></ul>(b) Show that this tangent is also a tangent to the circle with equation To prove tangency sub in equation of line into equation of circle and show b 2 – 4ac = 0.
  7. 7. <ul><li>4. Proving tangency / Common tangents </li></ul>8 Marks <ul><li>Show that the line x + 2y = 3 is a tangent to the circle </li></ul>(b) Show that this tangent is also a tangent to the circle with equation To prove tangency sub on equation of line into equation of circle and show b 2 – 4ac = 0.
  8. 8. <ul><li>4. Proving tangency / Common tangents </li></ul>8 Marks <ul><li>Show that the line x + 2y = 3 is a tangent to the circle </li></ul>(b) Show that this tangent is also a tangent to the circle with equation In part (b) do exactly the same!!!!!!
  9. 9. <ul><li>5. Points of intersection / Midpoint as centre </li></ul>To find points of intersection sub in line equation to circle equation So x = 6 or x = -2 A is x = 6 and so is (6, 6)
  10. 10. <ul><li>5. Points of intersection / Midpoint as centre </li></ul>Centre of circle will be midpoint of AB Radius is distance from centre to A or B Equation of circle:
  11. 11. <ul><li>5. Touching Circles </li></ul>4 Marks (a) (i) Show that the radius of circle P is (ii) Hence show that circles P and Q touch. Circle P has equation Circle Q has centre (-2, -1) and radius
  12. 12. 5. Touching Circles Solution to (a)(i): If given general form of circle’s equation remember 2g = -8 g = -4 2f = -10 f = -5
  13. 13. 5. Touching Circles Solution to (a)(ii): If circles touch distance between centres =sum of lengths of radii g = -4 f = -5 Centre of P is (4, 5)

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