First, find the derivative \(\frac{dy}{dx}\) of the curve \(y = 2 + \sqrt{25 - x^2}\).

Using the chain rule, \(\frac{dy}{dx} = \frac{1}{2}(25 - x^2)^{-1/2} \times (-2x)\).

Simplifying, \(\frac{dy}{dx} = \frac{-x}{\sqrt{25 - x^2}}\).

Set \(\frac{dy}{dx} = \frac{4}{3}\) and solve for \(x\):

\(\frac{-x}{\sqrt{25 - x^2}} = \frac{4}{3}\).

Square both sides: \(\frac{x^2}{25 - x^2} = \frac{16}{9}\).

Cross-multiply: \(16(25 - x^2) = 9x^2\).

Expand and simplify: \(400 - 16x^2 = 9x^2\).

Combine terms: \(25x^2 = 400\).

Solve for \(x\): \(x^2 = 16\), so \(x = \pm 4\).

Substitute \(x = -4\) into the original equation to find \(y\):

\(y = 2 + \sqrt{25 - (-4)^2} = 2 + \sqrt{9} = 2 + 3 = 5\).

Thus, the coordinates are \((-4, 5)\).