2018 UNCO Math Contest II Problems/Problem 3

Problem

Find all values of $B$ that have the property that if $(x, y)$ lies on the hyperbola $2y^2-x^2 = 1$ , then so does the point $(3x + 4y, 2x + By)$ .

Solution

We can write a system of equations - $2y^2-x^2 = 1$ $2(2x + By)^2 - (3x+4y)^2 = 1$

Expanding the second equation, we get $-x^2+8Bxy-24xy+2B^2y^2-16y^2=1$ .

Since we want this to look like $2y^2-x^2=1$ , we plug in B's that would put it into that form. If we plug in $B=3$ , things cancel, and we get $-x^2+24xy-24xy+18y^2-16y^2=1 \rightarrow 2y^2-x^2=1$ . So $\boxed{B=3}$ ~Ultraman

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
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2018 UNCO Math Contest II Problems/Problem 3

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