A laser spotlight on a robotic arm traces a path P P\,P on a high-precision sensor wall. The coordinates (x,y)(x, y)(x,y) of the spotlight at time θ \theta\,θ are given by the parametric equations
x=cosec θ,y=cot(θ+π6),π6<θ<π2 x = \text{cosec } \theta, \quad y = \cot \left( \theta + \frac{\pi}{6} \right), \quad \frac{\pi}{6} < \theta < \frac{\pi}{2} x=cosec θ,y=cot(θ+6π),6π<θ<2πFind dydx\displaystyle \frac{dy}{dx}dxdy in terms of θ\thetaθ.
Find an equation for the tangent to the path P P\,P at the point where θ=π3\displaystyle \theta = \frac{\pi}{3}θ=3π. Give your answer in the form y=mx+cy = mx + cy=mx+c, where m m\,m and c c\,c are constants.
Show that all points on the path P P\,P satisfy the equation
y=Ax2−Bx2−1x2−C y = \frac{A x^2 - B\sqrt{x^2 - 1}}{x^2 - C} y=x2−CAx2−Bx2−1where AAA, BBB, and C C\,C are constants to be determined.