2014 AMC 12B Problems/Problem 11

Revision as of 19:07, 20 February 2014 by Cnj333 (talk | contribs) (Solution)

Problem

Let $\mathcal P$ be the parabola with equation $y = x^2$ and let $Q = (20, 14)$. There are real numbers $r$ and $s$ such that the line through $Q$ with slope $m$ does not intersect $\mathcal P$ if and only if $r < m < s$. What is $r + s$?

\[\textbf{(A)}\ 1\qquad\textbf{(B)}\ 26\qquad\textbf{(C)}\ 40\qquad\textbf{(D)}}\ 52\qquad\textbf{(E)}\ 80\] (Error compiling LaTeX. Unknown error_msg)

Solution

The line through $Q$ only meets the parabola if the system of equations $y = x^2$ and $y -