2013 AIME II Problems/Problem 10

Problem 10

Given a circle of radius $\sqrt{13}$ , let $A$ be a point at a distance $4 + \sqrt{13}$ from the center $O$ of the circle. Let $B$ be the point on the circle nearest to point $A$ . A line passing through the point $A$ intersects the circle at points $K$ and $L$ . The maximum possible area for $\triangle BKL$ can be written in the form $\frac{a - b\sqrt{c}}{d}$ , where $a$ , $b$ , $c$ , and $d$ are positive integers, $a$ and $d$ are relatively prime, and $c$ is not divisible by the square of any prime. Find $a+b+c+d$ .

Solution 1

$[asy] import math; import olympiad; import graph; pair A, B, K, L; B = (sqrt(13), 0); A=(4+sqrt(13), 0); dot(B); dot(A); draw(Circle((0,0), sqrt(13))); label("$O$", (0,0), S);label("$B$", B, SW);label("$A$", A, S); dot((0,0)); [/asy]$

Draw $OC$ perpendicular to $KL$ at $C$ . Draw $BD$ perpendicular to $KL$ at $D$ .

$\frac{\triangle OKL}{\triangle BKL} = \frac{OC}{BD} = \frac{AO}{AB} = \frac{4+\sqrt{13}}{4}$

Therefore, to maximize area of $\triangle BKL$ , we need to maximize area of $\triangle OKL$ .

$\triangle OKL = \frac12 r^2 \sin{\angle KOL}$

So when area of $\triangle OKL$ is maximized, $\angle KOL = \frac{\pi}{2}$ .

Eventually, we get $\triangle BKL= (\frac12 \cdot \sqrt{13}^2)\cdot(\frac{4}{4+\sqrt{13}})=\frac{104-26\sqrt{13}}{3}$

So the answer is $104+26+13+3=\boxed{146}$ .

2013 AIME II (Problems • Answer Key • Resources)
Preceded by Problem 9	Followed by Problem 11
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2013 AIME II Problems/Problem 10

Problem 10

Solution 1

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