Difference between revisions of "2014 AMC 12B Problems/Problem 17"
Kevin38017 (talk | contribs) (Created page with "==Problem 17== Let <math>P</math> be the parabola with equation <math>y=x^2</math> and let <math>Q = (20, 14)</math>. There are real numbers <math>r</math> and <math>s</math> su...") |
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==Problem 17== | ==Problem 17== | ||
− | Let <math>P</math> be the parabola with equation <math>y=x^2</math> and let <math>Q = (20, 14)</math>. There are real numbers <math>r</math> and <math>s</math> such that the line through <math>Q</math> with slope <math>m</math> does not intersect <math>P</math> if and only if <math>r < m < s</math>. What is <math>r + s</math>? | + | Let <math>P</math> be the parabola with equation <math>y=x^2</math> and let <math>Q = (20, 14)</math>. There are real numbers <math>r</math> and <math>s</math> such that the line through <math>Q</math> with slope <math>m</math> does not intersect <math>P</math> if and only if <math>r</math> < <math>m</math> < <math>s</math>. What is <math>r + s</math>? |
<math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 26\qquad\textbf{(C)}\ 40\qquad\textbf{(D)}}\ 52\qquad\textbf{(E)}\ 80 </math> | <math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 26\qquad\textbf{(C)}\ 40\qquad\textbf{(D)}}\ 52\qquad\textbf{(E)}\ 80 </math> |
Revision as of 18:45, 20 February 2014
Problem 17
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 (Calculus-based)
The line will begin to intercept the parabola when its slope equals that of the parabola at the point of tangency. Taking the derivative of the equation of the parabola, we get that the slope equals . Using the slope formula, we find that the slope of the tangent line to the parabola also equals
. Setting these two equal to each other, we get
Solving for
, we get
The sum of the two possible values for
where the line is tangent to the parabola is
, and the sum of the slopes of these two tangent lines is equal to
, or
.
(Solution by kevin38017)