# Difference between revisions of "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