A precision-engineered micro-shuttle follows a path CCC in a magnetic field. The position of the shuttle at time ttt, where ttt is a parameter in radians, is given by the parametric equations
x=4cos2t,y=8sin3t,−π2<t<π2 x = 4 \cos 2t, \quad y = 8 \sin^3 t, \quad -\frac{\pi}{2} < t < \frac{\pi}{2} x=4cos2t,y=8sin3t,−2π<t<2πThe shuttle passes through point PPP when t=π6t = \frac{\pi}{6}t=6π.
The line lll represents the tangent to the shuttle's path at point PPP.
Use parametric differentiation to show that (i) dydx=ksint\frac{\mathrm{d}y}{\mathrm{d}x} = k \sin tdxdy=ksint where kkk is a constant to be found. (ii) an equation for the tangent line lll is 3x+4y−10=03x + 4y - 10 = 03x+4y−10=0.
The path CCC is intersected again by the line lll at the point QQQ.
Using algebra and showing detailed reasoning, find the exact coordinates of QQQ.
