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

As y approaches zero, the expression $-\frac{x}{4y}$ will approach negative or positive infinity, indicating vertical slopes along the x-axis. This means that at points where y is very small, the direction will change drastically, suggesting a reduction in the slope in the vicinity of the x-axis.

For positive values of y, the direction is downward when x is positive and upward when x is negative. For negative values of y, the behavior is reversed: downward when x is negative and upward when x is positive. This establishes a pattern of direction in the field.

Considering all the signs and behaviors from the analysis above, the representation that matches these conditions must be identified. The correct answer is A, which aligns with the deduction of slopes determined earlier.