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:

$\textbf{(A)}\ 2\qquad \textbf{(B)}\ 2\text{ or }3\qquad \textbf{(C)}\ 2\text{ or }4\qquad \textbf{(D)}\ 3\text{ or }4\qquad \textbf{(E)}\ 2,3,\text{ or }4$

Solution

Consider the hyperbola $x^2-y^2=1$ . 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. $y=x+3,\, x=0.). If the second line is instead$ x=4 $, it intersects the hyperbola twice, so the lines can intersect the hyperbola 3 times. Finally, if both lines intersect the hyperbola twice, such as$ y=2x-4 $and$ y=3x-6 $, the lines can intersect the hyperbola 4 times. So the answer is$ \textbf{(E)}\ 2,3,\text{ or } 4$

1956 AHSME (Problems • Answer Key • Resources)
Preceded by Problem 21	Followed by Problem 22
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1956 AHSME Problems/Problem 21

Problem 21

Solution

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