Difference between revisions of "1967 AHSME Problems/Problem 14"
(Created page with "== Problem == The number of distinct points common to the curves <math>x^2+4y^2=1</math> and <math>4x^2+y^2=4</math> is: <math>\text{(A) } 0 \quad \text{(B) } 1 \quad \text{(C) ...") |
(→Solution) |
||
Line 5: | Line 5: | ||
== Solution == | == Solution == | ||
− | <math>\fbox{C}</math> | + | Let <math>x^2 = a</math> and <math>y^2 = b.</math> |
+ | |||
+ | then | ||
+ | |||
+ | <math>a+4b=1</math> | ||
+ | |||
+ | and | ||
+ | |||
+ | <math>4a+b=4</math> | ||
+ | |||
+ | By solving we find-- | ||
+ | |||
+ | <math>a=1</math> | ||
+ | |||
+ | <math>b=0</math> | ||
+ | |||
+ | However <cmath>a=x^2</cmath> | ||
+ | and <cmath>b=y^2</cmath> | ||
+ | |||
+ | Therefore | ||
+ | <math>y=0</math>, and <math>x=1,-1</math> | ||
+ | |||
+ | Thus, the only solutions are <math>(0,1)</math>, and <math>(0,-1)</math> | ||
+ | |||
+ | |||
+ | So there are only 2 solutions | ||
+ | |||
+ | |||
+ | <math>=></math> <math>\fbox{C}</math> | ||
== See also == | == See also == |
Revision as of 17:19, 8 October 2015
Problem
The number of distinct points common to the curves and is:
Solution
Let and
then
and
By solving we find--
However and
Therefore , and
Thus, the only solutions are , and
So there are only 2 solutions
See also
1967 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.