The path of a beam of light reflecting off a parabolic mirror is modeled by the parametric equations
x=2.5t2 and y=5t,−4≤t≤4 x = 2.5t^2 \text{ and } y = 5t, \quad -4 \le t \le 4 x=2.5t2 and y=5t,−4≤t≤4where x x\,x and y y\,y are measured in centimeters.
Determine the Cartesian equation of the reflective surface in the form y2=f(x)y^2 = f(x)y2=f(x).
A specific photon strikes the mirror at point AAA where the parameter t=at = at=a (with a>0,a≠1a > 0, a \neq 1a>0,a=1). The tangent to the curve at point AAA makes an angle θ\thetaθ with a line through AAA parallel to the xxx-axis. The point BBB is located at (2.5,0)(2.5, 0)(2.5,0). The line segment ABABAB makes an angle ϕ\phiϕ with the positive xxx-axis.
By calculating the gradient of the curve, show that tanθ=1a\tan \theta = \frac{1}{a}tanθ=a1.
Find an expression for tanϕ\tan \phitanϕ in terms of aaa, simplifying your answer.
Hence, show that tan2θ=tanϕ\tan 2\theta = \tan \phitan2θ=tanϕ.
717 exam-style questions on Edexcel A Level Maths Differentiation, 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. Each one has a worked solution and a mark scheme showing where the marks go.