Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 4.8 AIME Prove it Philippines

Problem:
A line passes through $(k, -9)$ and $(7, 3k)$ and has slope $2k$ . Find the possible values of $k$ .

Solution

Solution:
We have $\frac{3k - (-9)}{7 - k} = 2k$ , so $3k + 9 = 2k(7 - k)$ or $2k^2 - 11k + 9 = (2k - 9)(k - 1) = 0$ . Hence $k = \frac{9}{2}$ or $k = 1$ .

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