A parabolic acoustic mirror for a long-range microphone is modeled by the parametric equations
x=5t2 and y=10t,−3≤t≤3 x = 5t^2 \text{ and } y = 10t, \quad -3 \le t \le 3 x=5t2 and y=10t,−3≤t≤3Determine a Cartesian equation for the profile of the mirror in the form y2=f(x)y^2 = f(x)y2=f(x).
A sound sensor is located at point B(5,0)B(5, 0)B(5,0). A specific point AAA on the mirror's profile corresponds to the parameter t=at = at=a, where a>1a > 1a>1. The tangent to the mirror at AAA makes an angle θ\thetaθ with a line through AAA parallel to the xxx-axis. The line segment ABABAB makes an angle ϕ\phiϕ with the positive xxx-axis.
(i) By finding an expression for dydx\frac{dy}{dx}dxdy in terms of ttt, show that tanθ=1a\tan \theta = \frac{1}{a}tanθ=a1.
(ii) Find tanϕ\tan \phitanϕ in terms of aaa.
(iii) Hence, prove that tan2θ=tanϕ\tan 2\theta = \tan \phitan2θ=tanϕ.
