# 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 $\textbf{(E)}\ 2,3,\text{ or } 4$

## See Also

 1956 AHSME (Problems • Answer Key • Resources) Preceded byProblem 21 Followed byProblem 22 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 • 26 • 27 • 28 • 29 • 30 All AHSME Problems and Solutions

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