Problem 25 · 2018 Math Kangaroo
Stretch
Algebra & Patterns
sum-constraint
A quadratic function of the form \(f(x) = x^2 + px + q\) intersects the x-axis and the y-axis in three different points. The circle through these three points intersects the graph of \(f\) in a fourth point. What are the coordinates of this fourth point of intersection?
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Answer: C — \((-p \mid q)\)
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Hint 1 of 2
The four intersection x-values of circle and parabola are the roots of one quartic.
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Hint 2 of 2
Use the sum of those roots; three of them you already know.
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Approach: sum of roots of the circle-parabola quartic
- Substituting y = x²+px+q into the circle equation gives a quartic in x whose four roots sum to −2p.
- Three known intersection x-values are the two parabola roots (summing to −p) and 0.
- So the fourth x-value is −2p −(−p) = −p, and y = f(−p) = q.
- The fourth point is (−p | q).
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