Mathematics

When you shift a coordinate system to a new origin, you're essentially substituting $x = X - 4$ and $y = Y + 1$ (where capital letters are the new coordinates). The key insight is that this translation eliminates the linear terms and reveals the "pure" quadratic structure of the curve. After substitution and simplification, you'll get an equation in $X$ and $Y$ with no first-degree terms—compare its coefficients to the target form $aX^2 + bXY + 1 = 0$ to find $a$ and $b$, then compute $b^2 - a$.

When you shift a coordinate system to a new origin, you're essentially substituting $x = X - 4$ and $y = Y + 1$ (where capital letters are the new coordinates). The key insight is that this translation eliminates the linear terms and reveals the "pure" quadratic structure of the curve. After substitution and simplification, you'll get an equation in $X$ and $Y$ with no first-degree terms—compare its coefficients to the target form $aX^2 + bXY + 1 = 0$ to find $a$ and $b$, then compute $b^2 - a$.