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Suppose a circle passes through (2,2) and (9,9) and touches the X - axis at P . If O is the origin then OP=

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3 years ago

```							Write the general equation of a circle with center (h,k) and radius r: (x - h)² + (y - k)² = r² We know 3 points on this circle, the two points given and the tangent point on the x-axis: (2,2) (9,9) (h,0) Plug in each of these points: (2 - h)² + (2 - k)² = r² (9 - h)² + (9 - k)² = r² (h - h)² + (0 - k)² = r² We get k² = r² from the last equation, so let's replace that in the first two equations: (2 - h)² + (2 - k)² = k² (9 - h)² + (9 - k)² = k² Expand both out: 4 - 4h + h² + 4 - 4k + k² = k² 81 - 18h + h² + 81 - 18k + k² = k² Cancel k²: 4 - 4h + h² + 4 - 4k = 0 81 - 18h + h² + 81 - 18k = 0 Simplify: h² - 4h + 8 - 4k = 0 h² - 18h + 162 - 18k = 0 We want to get rid of k, so multiply the first equation by 9 and the second equation by 2: 9h² - 36h + 72 - 36k = 0 2h² - 36h + 324 - 36k = 0 Subtract the second equation from the first: 7h² - 252 = 0 Divide both sides by 7: h² - 36 = 0 h² = 36 h = ±√36 But we only care about the positive solution since the negative wouldn't make sense. Answer: h = 6
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3 years ago
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