Which of the following best represents the direction field for the differential equation $$\frac{dy}{dx} = -\frac{x}{4y}$$ A - HSC - SSCE Mathematics Extension 1 - Question 7 - 2020 - Paper 1

To analyze the direction field, we note that it indicates the slope of the solution at each point in the $(x, y)$ plane. Specifically, the slope is negative for positive values of $y$ and positive for negative values of $y$. This suggests that as $x$ increases, $y$ will decrease when $y > 0$ and increase when $y < 0$.

Options B, C, and D can be ruled out because they do not conform to the behavior described above: they either have the wrong orientation or create inconsistencies in the expected slope behavior.

The correct representation of the direction field that matches the behavior of the solution to the given differential equation is option A.