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)
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== Solution ==
 
== Solution ==
<math>\fbox{C}</math>
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Let <math>x^2 = a</math> and <math>y^2 = b.</math>
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then
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        <math>a+4b=1</math>
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and
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        <math>4a+b=4</math>
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By solving we find--
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<math>a=1</math>
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<math>b=0</math>
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However <cmath>a=x^2</cmath>
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and <cmath>b=y^2</cmath>
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Therefore
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<math>y=0</math>, and <math>x=1,-1</math>
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Thus, the only solutions are <math>(0,1)</math>, and <math>(0,-1)</math>
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So there are only 2 solutions
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<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 $x^2+4y^2=1$ and $4x^2+y^2=4$ is:

$\text{(A) } 0 \quad \text{(B) } 1 \quad \text{(C) } 2 \quad \text{(D) } 3 \quad \text{(E) } 4$

Solution

Let $x^2 = a$ and $y^2 = b.$

then

        $a+4b=1$

and

        $4a+b=4$

By solving we find--

$a=1$

$b=0$

However \[a=x^2\] and \[b=y^2\]

Therefore $y=0$, and $x=1,-1$

Thus, the only solutions are $(0,1)$, and $(0,-1)$


So there are only 2 solutions


$=>$ $\fbox{C}$

See also

1967 AHSME (ProblemsAnswer KeyResources)
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

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