Difference between revisions of "2018 UNCO Math Contest II Problems/Problem 3"
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== 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> | ||
− | <math>B=3</math> | + | 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 == |
Revision as of 13:51, 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 |