1. Understanding the Problem and Key Concept

We are tasked with finding a specific point, let's call it $P(x_0, y_0)$, on the hyperbola $3x^2 - 4y^2 = 36$. This point must be the one closest to the given straight line $3x + 2y = 1$. Once we find $x_0$ and $y_0$, our final goal is to compute the value of the expression $\sqrt{2}(y_0 - x_0)$.

Key Concept: Point of Closest Approach For a point on a curve to be closest (or farthest) to a given straight line, the tangent to the curve at that point must be parallel to the