The path of a light ray reflected off a curved mirror is modeled by the equation
x=5sec22y,x>5,0<y<π4 x = 5\sec^2 2y, \quad x > 5, \quad 0 < y < \frac{\pi}{4} x=5sec22y,x>5,0<y<4πFind dxdy\displaystyle \frac{\text{d}x}{\text{d}y}dydx in terms of yyy.
Hence show that
dydx=pqxx−5 \frac{\text{d}y}{\text{d}x} = \frac{p}{qx\sqrt{x-5}} dxdy=qxx−5pwhere p p\,p is irrational and q q\,q is an integer, stating the values of p p\,p and qqq.
Find the equation of the normal to the path at the point where y=π12\displaystyle y = \frac{\pi}{12}y=12π, giving your answer in the form y=mx+cy = mx + cy=mx+c where m m\,m and c c\,c are exact constants.
