Differentiation

A robotic welding head follows a path $C\,$ in the $x y$ -plane described by the parametric equations

2\sin \theta + 5\cos \theta, \quad y = 4\cos^2 \theta + 2\sin \theta, \quad 0 \le \theta \le \pi

Show that $dydx=1\displaystyle \frac{dy}{dx} = 1$ where $θ=0\theta = 0$ .

[3]

The point $P\,$ lies on $C\,$ where $θ=0\theta = 0$ .

Find the equation of the tangent to the path $C\,$ at the point $P$ , giving your answer in the form $y = m x + c$ .

[3]

The tangent to the path at $P\,$ intersects the curve $C\,$ again at the point $Q$ .

Show that the value of $\theta\,$ at point $Q\,$ satisfies the equation

4\cos^2 \theta - 5\cos \theta + 1 = 0

[3]

Hence find the exact value of the $y$ -coordinate of $Q$ .

[3]