Circle tactics
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1. The document provides information on circle geometry, including how to find the equation of a circle given its center and radius, or given three points on the circle. 2. It discusses circle terminology such as concentric circles, congruent circles, and diametrically opposite points. 3. Steps are presented for using common sense to find diametrically opposite points, finding the equation of a circle from three points on the circle, and determining the closest distance between two circles.
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Here are some things you did well and could improve on:
WWW:
- You explained the key concepts around writing the equation of a circle clearly and concisely. Breaking it down step-by-step makes it easy to understand.
- Providing examples with worked solutions is very helpful for reinforcement. The visual diagrams additionally aid comprehension.
- Giving practice problems for students to try on their own, along with answers, allows for application of the material.
EBI:
- Some of the text could be formatted for easier reading (e.g. consistent formatting of equations).
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Circle tactics
- 2. Basic Tactics If given an equation find Centre Radius Draw diagrams Straight line stuff vital If finding equation need Centre Radius esp. dist formula x2 + y2 + 2gx + 2fy + c = 0 (x – a)2 + (y – b)2 = r2
- 3. Terminology Concentric circles? Same centre Congruent Same shape and size Diametrically Opposite Opposite ends of diameter A B
- 4. Circle Tactics If given an equation find Centre Radius Draw diagrams Straight line stuff vital If finding equation need Centre Radius using x2 + y2 + 2gx + 2fy + c = 0 (unless equation given with brackets) then use (x – a)2 + (y – b)2 = r2 - especially distance formula
- 5. Terminology Concentric circles? Same centre Congruent Same shape and size Diametrically Opposite Opposite ends of diameter
- 6. Using Common Sense! A(1 , 4) and B are diametrically opposite Centre is (6 , 10) B? (1,4) B (6,10) use steps 5 6 5 6 (11,16)
- 7. Using Common Sense! A(2 , 6) and B are diametrically opposite Centre is (5 , 13) B? (2,6) B (5,13) use steps 3 7 3 7 (8,20)
- 8. Reminder Perpendicular Bisector cut line in half at right angles x1+x2 y1+y2 2 2 m1m2 = -1
- 9. Finding Equation From Points On Circle If we have three points on a circle A,B and C then perpendicular bisectors will meet in the centre. We can use this to find the equation of a circle
- 10. A (0 , 2) B(1 , 5) and C(4 , 4) lie on a circle. Find its equation. Radius A B C (0 , 2) (1 , 5) (4 , 4) (½, 7 /2) m1 = 3 m2 = -1 /3 3y = -x + 11 (5 /2 , 9 /2) m1 = -1 /3 m2 = 3 y = 3x – 3 Lines meet at (2 , 3) (2 , 3) √5 (x – 2)2 + (y – 3)2 = 5
- 11. Finding Equation From Points On Circle If we have three points on a circle A,B and C then perpendicular bisectors will meet in the centre. We can use this to find the equation of a circle
- 12. Equation from 3 Points Perpendicular Bisectors m1m2 = -1 mid pt Centre – Point of Intersection Radius Distance Formula (y = y or sim. equations)
- 13. Closest Distance Find the closest distance between circle A x2 + y2 – 6x – 2y + 9 = 0 and B x2 + y2 – 14x – 8y + 61 = 0 (3 , 1) (7 , 4) 1 2 2 A B 5
- 14. Closest Distance Closest distance between 2 circles (given equations) ? Dist between centres – (sum of radii) Get centres
- 15. Proving Circles Touch Externally Prove these touch externally A x2 + y2 – 6x – 4y – 23 = 0 B x2 + y2 – 18x – 20y + 165 = 0 (3 , 2) (9 , 10) A B 6 4 10
- 16. Proving Circles Touch Internally Prove these touch internally A x2 + y2 – 4x – 6y – 51 = 0 B x2 + y2 – 12x – 12y + 63 = 0 (2 , 3) (6 , 6) 8 3 5
- 17. Proving circles touch at one point Externally Dist between centres = Sum of Radii
- 18. Proving Circles Touch Distance between centres = difference between radii Internally