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Interval of a Slope of a Line to not Intersect a Parabola

Source: AMC 12B #17

February 20, 2014
analytic geometrygraphing linesslopeconicsparabolaquadraticscalculus

Problem Statement

Let PP be the parabola with equation y=x2y = x^2 and let Q=(20,14)Q = (20, 14) There are real numbers rr and ss such that the line through QQ with slope mm does not intersect PP if and only if r<m<sr < m < s. What is r+s?r + s?
<spanclass=latexbold>(A)</span>1<spanclass=latexbold>(B)</span>26<spanclass=latexbold>(C)</span>40<spanclass=latexbold>(D)</span>52<spanclass=latexbold>(E)</span>80 <span class='latex-bold'>(A)</span> 1 \qquad <span class='latex-bold'>(B)</span> 26 \qquad <span class='latex-bold'>(C)</span> 40 \qquad <span class='latex-bold'>(D)</span> 52 \qquad <span class='latex-bold'>(E)</span> 80 \qquad