Difference between revisions of "2014 AMC 12B Problems/Problem 17"
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So <math>m < 0</math> for <math>r < m < s</math> where <math>r</math> and <math>s</math> are the roots of <math>m^2 - 80m + 56 = 0</math> and their sum by Vieta's formulas is <math>\boxed{\textbf{(E)}\ 80}</math>. | So <math>m < 0</math> for <math>r < m < s</math> where <math>r</math> and <math>s</math> are the roots of <math>m^2 - 80m + 56 = 0</math> and their sum by Vieta's formulas is <math>\boxed{\textbf{(E)}\ 80}</math>. | ||
− | ==Solution (Calculus-based)== | + | ==Solution 2 (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 <math>2x</math>. Using the slope formula, we find that the slope of the tangent line to the parabola also equals <math>\frac{14-x^2}{20-x}</math>. Setting these two equal to each other, we get | 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 <math>2x</math>. Using the slope formula, we find that the slope of the tangent line to the parabola also equals <math>\frac{14-x^2}{20-x}</math>. Setting these two equal to each other, we get |
Revision as of 19:17, 17 April 2020
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
?
Solution (Algebra Based)
Let . Equating them:
For there to be no solutions, the discriminant must be less than zero:
.
So for
where
and
are the roots of
and their sum by Vieta's formulas is
.
Solution 2 (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
.
See also
2014 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 16 |
Followed by Problem 18 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
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