A small robot moves on a flat floor. Its position is (x, y) in metres…

A small robot moves on a flat floor. Its position is $(x, y)$ in metres. The controller turns the robot so that at every point its path has the slope $\dfrac{dy}{dx} = -\dfrac{x}{y}$ for $y > 0$. The robot starts at the point $(3\ \text{m}, 4\ \text{m})$. By separating the variables, the designer writes $y\,dy = -x\,dx$ and integrates. She finds that the path is part of a circle centred at the origin. Find the radius of the circle in metres. Then find the height $y$ of the path when $x = -4\ \text{m}$, still on the upper half of the circle. Find the slope of the path there, to four decimal places.