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.
Practise Edexcel A Level Maths Differentiation with exam-style questions for A Level Maths. 311 questions covering 9.1 Differentiating sin x and cos x, 9.2 Differentiating exponentials and logarithms, 9.3 The Chain Rule, 9.4 The Product Rule, 9.5 The Quotient Rule, 9.6 Differentiating Trigonometric Functions, 9.7 Parametric Differentiation, 9.8 Implicit Differentiation, 9.9 Using Second Derivatives, and 9.10 Rates of Change, matched to the Edexcel A Level Maths (9MA0) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.
