Difference between revisions of "1956 AHSME Problems/Problem 21"
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==Solution== | ==Solution== | ||
Consider the hyperbola <math>x^2-y^2=1</math>. | Consider the hyperbola <math>x^2-y^2=1</math>. | ||
− | 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. <math>y=x+3,\, x=0.). If the second line is instead < | + | 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. <math>y=x+3,\, x=0.</math>). If the second line is instead <math>x=4</math>, it intersects the hyperbola twice, so the lines can intersect the hyperbola 3 times. Finally, if both lines intersect the hyperbola twice, such as <math>y=2x-4</math> and <math>y=3x-6</math>, the lines can intersect the hyperbola 4 times. So the answer is <math>\textbf{(E)}\ 2,3,\text{ or } 4</math> |
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==See Also== | ==See Also== | ||
{{AHSME box|year=1956|num-b=21|num-a=22}} | {{AHSME box|year=1956|num-b=21|num-a=22}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 13:27, 30 April 2017
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) | ||
Preceded by Problem 21 |
Followed by Problem 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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