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Understanding the Circle Equation and Distance Formula

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This handout explains the distance formula for points in a plane and introduces the circle equation. The distance between points (a, b) and (x, y) is derived using the Pythagorean theorem as d² = (x-a)² + (y-b)². The equation of a circle centered at (a, b) with radius r is given by (x-a)² + (y-b)² = r². Several examples illustrate how to calculate the center and radius from given equations, including instances where the center is at the origin, simplifying the circle equation.

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Understanding the Circle Equation and Distance Formula

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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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