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CIRCLES

CIRCLES. Distance Formula (see handout). The distance from (a,b) to (x,y) is given by d 2 = (x-a) 2 + (y-b) 2. (x,y). Proof. d. (y – b). (a,b). (x,b). (x – a). By Pythagoras. d 2 = (x-a) 2 + (y-b) 2. The Circle Equation (x-a) 2 + (y-b) 2 = r 2.

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CIRCLES

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  1. CIRCLES Distance Formula (see handout) The distance from (a,b) to (x,y) is given by d2 = (x-a)2 + (y-b)2 (x,y) Proof d (y – b) (a,b) (x,b) (x – a) By Pythagoras d2 = (x-a)2 + (y-b)2

  2. The Circle Equation (x-a)2 + (y-b)2 = r2 A circle with centre (a,b) and radius , r, has equation (x-a)2 + (y-b)2 = r2 Proof Suppose P(x,y) is any point on the circumference of a circle with centre A(a,b) and radius ,r. ie Y P(x,y) AP = r r So AP2 = r2 A(a,b) applying distance formula (x-a)2 + (y-b)2 = r2 X

  3. Ex1 (x-2)2 + (y-5)2 = 49 centre (2,5) radius = 7 (x+5)2 + (y-1)2 = 13 centre (-5,1) radius = 13 (x-3)2 + y2 = 20 centre (3,0) radius = 20 = 4 X 5 = 25 Ex2 Centre (2,-3) & radius = 10 NAB Equation is(x-2)2 + (y+3)2 = 100 Ex3 Centre (0,6) & radius = 23 r2 =23 X 23 Equation isx2 + (y-6)2 = 12 =49 = 12

  4. The Circle Equation x2 + y2 = r2 Note: a simpler version of the first equation is obtained whenever (a,b) = (0,0) ie centre is at origin. (x-a)2 + (y-b)2 = r2 becomes (x-0)2 + (y-0)2 = r2 or x2 + y2 = r2 Ex4 x2 + y2 = 7 has centre (0,0) & radius = 7 Ex5 The circle with centre (0,0) & radius = 1/3 has equation x2 + y2 = 1/9

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