Difference between revisions of "1957 AHSME Problems/Problem 34"
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== Solution == | == Solution == | ||
| − | <math>\fbox{\textbf{(C)} a straight line segment not including the end-points}</math>. | + | |
| + | <asy> | ||
| + | |||
| + | import geometry; | ||
| + | |||
| + | // Axes | ||
| + | draw((0,-8)--(0,8),arrow=Arrows); | ||
| + | label("$y$",(0,9)); | ||
| + | draw((-8,0)--(8,0),arrow=Arrows); | ||
| + | label("$x$",(9,0)); | ||
| + | |||
| + | // Circle and Line | ||
| + | draw(circle((0,0),5),red+dashed); | ||
| + | draw(line((0,1),(1,0)),blue); | ||
| + | |||
| + | </asy> | ||
| + | |||
| + | The line <math>x+y=1</math> is in [[standard form]]. Thus, it has a slope of <math>-\tfrac{1}{1}=-1</math> and a <math>y</math>-intercept at <math>(0,1)</math>. It is graphed in blue in the diagram. All the points that satisfy this system must lie on this line. | ||
| + | |||
| + | The inequality <math>x^2+y^2<25</math> is true for all points [[strict inequality|strictly]] less than [[distance formula|distance]] <math>\sqrt{25}=5</math> from the origin. This area is a [[disk]] with radius <math>5</math>, whose open boundary is represented in red in the diagram. | ||
| + | |||
| + | Because the <math>y</math>-intercept of the line, <math>(0,1)</math>, is inside the disk, it cannot be tangent to the disk. Thus, there must be infiniely many points along the line which are within the disk and thereby satisfy the system. However, because the inequality is strict, the segment of solutions does not include the endpoints, so our answer is <math>\fbox{\textbf{(C)} a straight line segment not including the end-points}</math>. | ||
== See Also == | == See Also == | ||
| − | {{AHSME 50p box|year=1957 | + | {{AHSME 50p box|year=1957|num-b=33|num-a=35}} |
{{MAA Notice}} | {{MAA Notice}} | ||
[[Category:AHSME]][[Category:AHSME Problems]] | [[Category:AHSME]][[Category:AHSME Problems]] | ||
| + | [[Category:Introductory Algebra Problems]] | ||
Latest revision as of 07:25, 26 July 2024
Problem
The points that satisfy the system 
, constitute the following set:
Solution
![[asy] import geometry; // Axes draw((0,-8)--(0,8),arrow=Arrows); label("$y$",(0,9)); draw((-8,0)--(8,0),arrow=Arrows); label("$x$",(9,0)); // Circle and Line draw(circle((0,0),5),red+dashed); draw(line((0,1),(1,0)),blue); [/asy]](http://latex.artofproblemsolving.com/b/8/d/b8d0c917ef633c50269b9ae3fab786f05a474f55.png)
The line 
is in standard form. Thus, it has a slope of 
and a 
-intercept at 
. It is graphed in blue in the diagram. All the points that satisfy this system must lie on this line.
The inequality 
is true for all points strictly less than distance 
from the origin. This area is a disk with radius 
, whose open boundary is represented in red in the diagram.
Because the -intercept of the line, , is inside the disk, it cannot be tangent to the disk. Thus, there must be infiniely many points along the line which are within the disk and thereby satisfy the system. However, because the inequality is strict, the segment of solutions does not include the endpoints, so our answer is 
.
See Also
| 1957 AHSC (Problems • Answer Key • Resources) | ||
| Preceded by Problem 33 |
Followed by Problem 35 | |
| 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 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50 | ||
| All AHSME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
