<p>The parabola with equation <img alt="$y = x^2 - 4$" height="20" src="https://latex.artofproblemsolving.com/1/c/a/1ca935bc10b4a49310308de10b1f5ef27d530102.png" width="85" /> is rotated <img alt="$60^\circ$" height="13" src="https://latex.artofproblemsolving.com/2/9/f/29fa0a9949f018e121b2093216dd5db13d3abef0.png" width="24" /> counterclockwise around the origin. The unique point in the fourth quadrant where the original parabola and its image intersect has <img alt="$y$" height="13" src="https://latex.artofproblemsolving.com/0/9/2/092e364e1d9d19ad5fffb0b46ef4cc7f2da02c1c.png" width="11" />-coordinate <img alt="$\frac{a - \sqrt{b}}{c}$" height="44" src="https://latex.artofproblemsolving.com/0/5/2/05226fad0479321c09b58b5eddd3f125c42489bc.png" width="61" />, where <img alt="$a$" height="10" src="https://latex.artofproblemsolving.com/c/7/d/c7d457e388298246adb06c587bccd419ea67f7e8.png" width="11" />, <img alt="$b$" height="14" src="https://latex.artofproblemsolving.com/8/1/3/8136a7ef6a03334a7246df9097e5bcc31ba33fd2.png" width="10" />, and <img alt="$c$" height="8" src="https://latex.artofproblemsolving.com/3/3/7/3372c1cb6d68cf97c2d231acc0b47b95a9ed04cc.png" width="8" /> are positive integers, and <img alt="$a$" height="10" src="https://latex.artofproblemsolving.com/c/7/d/c7d457e388298246adb06c587bccd419ea67f7e8.png" width="11" /> and <img alt="$c$" height="8" src="https://latex.artofproblemsolving.com/3/3/7/3372c1cb6d68cf97c2d231acc0b47b95a9ed04cc.png" width="8" /> are relatively prime. Find <img alt="$a + b + c$" height="14" src="https://latex.artofproblemsolving.com/2/c/d/2cdfeae33c1e09de6207d446beb72396c04be481.png" width="70" />.</p>

Question No: 406

The parabola with equation $y = x^2 - 4$ is rotated $60^\circ$ counterclockwise around the origin. The unique point in the fourth quadrant where the original parabola and its image intersect has $y$ -coordinate $\frac{a - \sqrt{b}}{c}$ , where $a$ , $b$ , and $c$ are positive integers, and $a$ and $c$ are relatively prime. Find $a + b + c$ .

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Tags: Algebra Geometry

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