2014 AMC 12B Problems/Problem 11
Problem
Let be the parabola with equation
and let
. There are real numbers
and
such that the line through
with slope
does not intersect
if and only if
. What is
?
\[\textbf{(A)}\ 1\qquad\textbf{(B)}\ 26\qquad\textbf{(C)}\ 40\qquad\textbf{(D)}}\ 52\qquad\textbf{(E)}\ 80\] (Error compiling LaTeX. Unknown error_msg)
Solution
The line through only meets the parabola if the system of equations
and
has one or more solution(s) (the second equation comes from point-slope form). We can equate
and get the equation
. Now if we move all the terms to the left, we get
. We want this equation to have no solutions, and are trying to find the bounds of
and
for
such that this is unsolvable. A quadratic is unsolvable across the reals if its discriminant is less than 0. So, we set the discriminant to be less than 0:
We can square each side, keeping track of extraneous solutions: the resulting equation is not just
, but
. The solutions to this equation are
and
: to get their sum we use Vieta's Formulas and get
. Because
is not a valid answer choice, we can know for certain that the correct answer is
.