# Difference between revisions of "2013 AIME II Problems/Problem 10"

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Then we get <math>(k^2+1)x^2-2k^2(4+\sqrt{13})x+k^2\cdot (4+\sqrt{13})^2-13=0</math>, according to Vieta's formulas, we get | Then we get <math>(k^2+1)x^2-2k^2(4+\sqrt{13})x+k^2\cdot (4+\sqrt{13})^2-13=0</math>, according to Vieta's formulas, we get | ||

− | <math> | + | <math>x_1+x_2=\frac{2k^2(4+\sqrt{13})}{k^2+1}</math>, and <math>x_1x_2=\frac{(4+\sqrt{13})^2\cdot k^2-13}{k^2+1}</math> |

− | So, <math>LK=\sqrt{1+k^2}\cdot \sqrt{( | + | So, <math>LK=\sqrt{1+k^2}\cdot \sqrt{(x_1+x_2)^2-4x_1x_2}</math> |

Also, the distance between <math>O</math> and <math>LK</math> is <math>\frac{k\times \sqrt{13}-(4+\sqrt{13})\cdot k}{\sqrt{1+k^2}}=\frac{-4k}{\sqrt{1+k^2}}</math> | Also, the distance between <math>O</math> and <math>LK</math> is <math>\frac{k\times \sqrt{13}-(4+\sqrt{13})\cdot k}{\sqrt{1+k^2}}=\frac{-4k}{\sqrt{1+k^2}}</math> | ||

− | So the ares <math>S=0.5ah=\frac{-4k\sqrt{(16-8\sqrt{13})k^2-13}}{k^2+1} | + | So the ares <math>S=0.5ah=\frac{-4k\sqrt{(16-8\sqrt{13})k^2-13}}{k^2+1}</math> |

− | Then the maximum value of < | + | Then the maximum value of <math>S</math> is <math>\frac{104-26\sqrt{13}}{3}</math> |

− | So the answer is < | + | So the answer is <math>104+26+13+3=\boxed{146}</math> |

## Revision as of 04:06, 5 April 2013

Given a circle of radius , let be a point at a distance from the center of the circle. Let be the point on the circle nearest to point . A line passing through the point intersects the circle at points and . The maximum possible area for can be written in the form , where , , , and are positive integers, and are relatively prime, and is not divisible by the square of any prime. Find .

## Solution

Now we put the figure in the Cartesian plane, let the center of the circle , then , and

The equation for Circle O is , and let the slope of the line be , then the equation for line is

Then we get , according to Vieta's formulas, we get

, and

So,

Also, the distance between and is

So the ares

Then the maximum value of is

So the answer is