A high-precision cam profile in a mechanical sensor follows a path defined by the equation

\sin 4\theta \quad 0 \le x \le 6, \quad 0 \le \theta \le \frac{\pi}{8}

where $x$ is the horizontal displacement in millimetres and $θ\theta$ is the angular position of the cam in radians.

Find $dxdθ\frac{dx}{d\theta}$ in terms of $θ\theta$ .

Hence show that

\frac{d\theta}{dx} = \frac{k}{\sqrt{36-x^2}}

where $k$ is a constant to be determined.

A specific calibration point $P (a, b)$ lies on the profile. At this point:

The rate of change of the angular position with respect to displacement, $dθdx\frac{d\theta}{dx}$ , is exactly $1123\frac{1}{12\sqrt{3}}$ .
Both $a$ and $b$ are positive constants.

Determine the exact values of $a$ and $b$ .