Difference between revisions of "2018 UNCO Math Contest II Problems/Problem 3"
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== Problem == | == Problem == | ||
+ | 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 <math>(3x + 4y, 2x + By)</math>. | ||
+ | == 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 -
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 |