A high-precision industrial laser tracks a path C C\,C on a component, defined by the parametric equations
x=3cos2t−6sint,y=6sint+5cost,0≤t≤2π x = 3\cos^2 t - 6\sin t, \quad y = 6\sin t + 5\cos t, \quad 0 \le t \le 2\pi x=3cos2t−6sint,y=6sint+5cost,0≤t≤2πShow that dydx=−1\displaystyle \frac{dy}{dx} = -1dxdy=−1 at the point where t=πt = \pit=π.
The point P P\,P lies on the path where t=πt = \pit=π.
Find the equation of the tangent to the laser's path at P P\,P in the form y=mx+cy = mx + cy=mx+c, where m m\,m and c c\,c are constants to be determined.
The laser's path is such that the tangent at P P\,P intersects the path C C\,C again at the point QQQ.
Show that the value of t t\,t at point Q Q\,Q satisfies the equation
3cos2t+5cost+2=0 3\cos^2 t + 5\cos t + 2 = 0 3cos2t+5cost+2=0Hence find the exact possible values of the yyy-coordinate of point QQQ.
