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y = x^2 + px + q, y = 2x +p are tangent (III Soros Olympiad 1996-97 R1 9.2)

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May 29, 2024
analytic geometrygeometryconics

Problem Statement

It is known that the graph of a quadratic trinomial y=x2+px+qy = x^2 + px + q touches the graph of a straight line y=2x+py = 2x + p. Prove that all such quadratic trinomials have the same minimum value. Find this smallest value.
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