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Problem 641

AIME late

Algebra Difficulty 5.1 Find the answer

5. The line $y=k x-2$ intersects the parabola $y^{2}=8 x$ at points $A, B$ . If the x-coordinate of the midpoint of segment $A B$ is 2, then the length of segment $A B$ $|A B|=$ $\qquad$

Your answer

Official solution

$\left\{\begin{array}{l} y=k x-2, \\ y^{2}=8 x \end{array} \Rightarrow k^{2} x^{2}-(4 k+8) x+4=0 \Rightarrow x_{0}=\frac{2 k+4}{k^{2}}=2 \Rightarrow k^{2}=k+2\right.$
$\Rightarrow k=2 \Rightarrow x^{2}-4 x+1=0$ . So $|A B|=\sqrt{5} \cdot \sqrt{16-4}=2 \sqrt{15}$ .

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.