Difference between revisions of "2018 UNCO Math Contest II Problems/Problem 3"
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== Problem == | == Problem == | ||
| − | Find all values of B that have the property that if (x, y) lies on the hyperbola 2y^2-x^2 = 1, | + | Find all values of <math>B</math> that have the property that if <math>(x, y)</math> lies on the hyperbola <math>2y^2-x^2 = 1</math>, |
| − | then so does the point (3x + 4y, 2x + By). | + | then so does the point <math>(3x + 4y, 2x + By)</math>. |
== Solution == | == Solution == | ||
| + | We can write a system of equations - | ||
| + | <cmath>2y^2-x^2 = 1</cmath> | ||
| + | <cmath>2(2x + By)^2 - (3x+4y)^2 = 1</cmath> | ||
| + | Expanding the second equation, we get <math>-x^2+8Bxy-24xy+2B^2y^2-16y^2=1</math>. | ||
| + | |||
| + | Since we want this to look like <math>2y^2-x^2=1</math>, we plug in B's that would put it into that form. If we plug in <math>B=3</math>, things cancel, and we get <math>-x^2+24xy-24xy+18y^2-16y^2=1 \rightarrow 2y^2-x^2=1</math>. So <math>\boxed{B=3}</math> | ||
| + | ~Ultraman | ||
== See also == | == See also == | ||
{{UNCO Math Contest box|year=2018|n=II|num-b=2|num-a=4}} | {{UNCO Math Contest box|year=2018|n=II|num-b=2|num-a=4}} | ||
| − | [[Category:]] | + | [[Category:Introductory Algebra Problems]] |
Revision as of 14:53, 2 January 2020
Problem
Find all values of 
that have the property that if 
lies on the hyperbola 
,
then so does the point 
.
Solution
We can write a system of equations -
![\[2y^2-x^2 = 1\]](http://latex.artofproblemsolving.com/8/f/b/8fb80c9751e3f20d5c76f1dd3d03b03f7ee165cd.png)
![\[2(2x + By)^2 - (3x+4y)^2 = 1\]](http://latex.artofproblemsolving.com/6/4/c/64c02b4eb643a2cc71645f62d83190f85330a521.png)
Expanding the second equation, we get 
.
Since we want this to look like 
, we plug in B's that would put it into that form. If we plug in 
, things cancel, and we get 
. So 
~Ultraman
See also
| 2018 UNCO Math Contest II (Problems • Answer Key • Resources) | ||
| Preceded by Problem 2 |
Followed by Problem 4 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 | ||
| All UNCO Math Contest Problems and Solutions | ||
