1956 AHSME Problems/Problem 21
Problem 21
If each of two intersecting lines intersects a hyperbola and neither line is tangent to the hyperbola, then the possible number of points of intersection with the hyperbola is:
Solution
Consider the hyperbola . It is possible for the two intersecting lines to intersect the hyperbola at 2 points if one of them has a slope of 1 and only intersects one part of the hyperbola and the other line doesn't intersect the hyperbola at all (Ex. ). If the second line is instead , it intersects the hyperbola twice, so the lines can intersect the hyperbola 3 times. Finally, if both lines intersect the hyperbola twice, such as and , the lines can intersect the hyperbola 4 times. So the answer is
See Also
1956 AHSME (Problems • Answer Key • Resources) | ||
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Followed by Problem 22 | |
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