2019/20 AHSMC Part 1 Problem 8

Problem 8 of 16

There are exactly two real values of $a$ such that the line $y = ax - 5$ intersects the parabola $y = x^2 + x + 1$ at exactly one point. If $a_1$ and $a_2$ are these values, then $a_1 \cdot a_2$ is equal to

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The line $y = ax - 5$ intersects the parabola $y = x^2 + x + 1$ at exactly one point if the equation $x^2 + x + 1 = ax - 5 \iff x^2 + (1-a)x + 6 = 0$ has only one solution. This happens if and only if $\Delta = (1-a)^2 - 24 = 0 \iff a_1 = 1 - \sqrt{24}$ , $a_2 = 1 + \sqrt{24}$ and thus $a_1 \cdot a_2 = -23$ . The answer is (B).

University of Alberta

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