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 ?
Answer Choices:
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Solution:
The line passing through point with slope has equation . The requested values for are those for which the system
has no solutions. Solving for in the first equation and substituting into the second yields , which reduces to . This equation has no solution for when the discriminant is negative, that is, when . This quadratic in is negative between its two roots , which are the required values of and . The requested sum is .
The problems on this page are the property of the MAA's American Mathematics Competitions