Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "1958 AHSME Problems/Problem 38"

(Created page with "== Problem == Let <math> r</math> be the distance from the origin to a point <math> P</math> with coordinates <math> x</math> and <math> y</math>. Designate the ratio <math> \fra...")
 
m (→‎Problem)
 
Line 1: Line 1:
 
== Problem ==
 
== Problem ==
−
Let <math> r</math> be the distance from the origin to a point <math> P</math> with coordinates <math> x</math> and <math> y</math>. Designate the ratio <math> \frac{y}{r}</math> by <math> s</math> and the ratio <math> \frac{x}{r}</math>  by <math> c</math>. Then the values of <math> s^2 \minus{} c^2</math> are limited to the numbers:
+
Let <math> r</math> be the distance from the origin to a point <math> P</math> with coordinates <math> x</math> and <math> y</math>. Designate the ratio <math> \frac{y}{r}</math> by <math> s</math> and the ratio <math> \frac{x}{r}</math>  by <math> c</math>. Then the values of <math> s^2 - c^2</math> are limited to the numbers:
  
−
<math> \textbf{(A)}\ \text{less than }{\minus{}1}\text{ are greater than }{\plus{}1}\text{, both excluded}\qquad\\
+
<math> \textbf{(A)}\ \text{less than }{-1}\text{ are greater than }{+1}\text{, both excluded}\qquad\\
−
\textbf{(B)}\ \text{less than }{\minus{}1}\text{ are greater than }{\plus{}1}\text{, both included}\qquad \\
+
\textbf{(B)}\ \text{less than }{-1}\text{ are greater than }{+1}\text{, both included}\qquad \\
−
\textbf{(C)}\ \text{between }{\minus{}1}\text{ and }{\plus{}1}\text{, both excluded}\qquad \\
+
\textbf{(C)}\ \text{between }{-1}\text{ and }{+1}\text{, both excluded}\qquad \\
−
\textbf{(D)}\ \text{between }{\minus{}1}\text{ and }{\plus{}1}\text{, both included}\qquad \\
+
\textbf{(D)}\ \text{between }{-1}\text{ and }{+1}\text{, both included}\qquad \\
−
\textbf{(E)}\ {\minus{}1}\text{ and }{\plus{}1}\text{ only}</math>
+
\textbf{(E)}\ {-1}\text{ and }{+1}\text{ only}</math>
  
 
== Solution ==
 
== Solution ==

Latest revision as of 23:23, 13 March 2015

Problem

Let $r$ be the distance from the origin to a point $P$ with coordinates $x$ and $y$. Designate the ratio $\frac{y}{r}$ by $s$ and the ratio $\frac{x}{r}$ by $c$. Then the values of $s^2 - c^2$ are limited to the numbers:

$\textbf{(A)}\ \text{less than }{-1}\text{ are greater than }{+1}\text{, both excluded}\qquad\\ \textbf{(B)}\ \text{less than }{-1}\text{ are greater than }{+1}\text{, both included}\qquad \\ \textbf{(C)}\ \text{between }{-1}\text{ and }{+1}\text{, both excluded}\qquad \\ \textbf{(D)}\ \text{between }{-1}\text{ and }{+1}\text{, both included}\qquad \\ \textbf{(E)}\ {-1}\text{ and }{+1}\text{ only}$

Solution

$\fbox{}$

See Also

1958 AHSC (Problems • Answer Key • Resources)
Preceded by
Problem 37 		Followed by
Problem 39
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. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1958_AHSME_Problems/Problem_38&oldid=68958"