Given the function \(f(x) = (4x + 2)^{-2}\), we need to find the derivative \(\frac{dy}{dx}\).

Using the chain rule, \(\frac{dy}{dx} = -2(4x + 2)^{-3} \times 4\).

This simplifies to \(\frac{dy}{dx} = -8(4x + 2)^{-3}\).

We know that the \(y\)-coordinate is decreasing at the rate of \(k\) and the \(x\)-coordinate is increasing at the rate of \(k\), so \(\frac{dy}{dx} = -1\).

Set \(-8(4x + 2)^{-3} = -1\).

Solving for \(x\), we get \((4x + 2)^{3} = 8\).

Taking the cube root, \(4x + 2 = 2\).

Solving for \(x\), \(4x = 0\) so \(x = 0\).

Substitute \(x = 0\) back into \(f(x)\) to find \(y\):

\(y = (4(0) + 2)^{-2} = 2^{-2} = \frac{1}{4}\).

Therefore, the coordinates of A are \((0, \frac{1}{4})\).