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The Distance Formula & Equations of Circles

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1 The Distance Formula & Equations of Circles
Proving the distance formula Proving equation of circles Examples of finding the distance between two points Examples of equations of circles

2 Proving the Distance Formula
If you have two points and are trying to find the distance between those points, you use the Pythagorean Theorem. So, using the Pythagorean Theorem (a2 + b2 = c2), you can assume that α2 + β2 = µ2 So, the distance from P to Q is written: d(P,Q) = (x1,y1) (x2,y2) (x2,y1) α β Q P Where α = x2-x1 andβ = y2-y1 and µ = distance from point P to point Q

3 Examples Find the distance between the points (2,4) & (3,-5)
sqrt[(3-2)2 + (-5-4)2] = sqrt(1 + 81) = sqrt(82) (2,4) (3,-5)

4 Examples cont. Find the distance between the points (-5,-5) & (4,-3)
sqrt[(-5-4)2 + (-5-(-3))2] = sqrt(81 + 4) = sqrt(85) (4,-3) (-5,-5)

5 Examples cont. Find the distance between the points (1,4) & (7,4)
sqrt[(1-7)2 + (4-4)-2] = sqrt(36 + 0) = sqrt(36) = 6 (7,4) (1,4)

6 Equation of Circles With a circle with a radius r, centered at point P, all points that are r units away from P make the circle. If the coordinates of P are (h,k), you can use the distance formula to get If we square this, we get the standard equation of a circle: (x-h)2 + (y-k)2 = r r P r (h,k)

7 Examples Draw the graph of the equation (x-2)2 + (y+2)2 = 9
(h,k) = (2,-2) r = 3

8 Examples cont. Draw the graph of the equation (x+3)2 + (y+4)2 = 4
(h,k) = (-3,-4) r = 2


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