1.5 Coordinate Geometry

A high-precision industrial laser tracks a path $C\,$ on a component, defined by the parametric equations

3\cos^2 t - 6\sin t, \quad y = 6\sin t + 5\cos t, \quad 0 \le t \le 2\pi

Show that $dydx=−1\displaystyle \frac{dy}{dx} = -1$ at the point where $\pi$ .

[2]

The point $P\,$ lies on the path where $\pi$ .

Find the equation of the tangent to the laser's path at $P\,$ in the form $y = m x + c$ , where $m\,$ and $c\,$ are constants to be determined.

[3]

The laser's path is such that the tangent at $P\,$ intersects the path $C\,$ again at the point $Q$ .

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

3\cos^2 t + 5\cos t + 2 = 0

[3]

Hence find the exact possible values of the $y$ -coordinate of point $Q$ .

[2]

1.5 Coordinate Geometry Questions