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.
178 exam-style questions on OCR (MEI) A Level Maths 1.5 Coordinate Geometry, covering 1.5.1 Equation of a straight line y = mx + c, 1.5.2 Gradients of parallel and perpendicular lines, 1.5.3 Distance between two points, 1.5.4 Midpoint of a line segment, 1.5.5 Form the equation of a straight line, 1.5.6 Draw a line given its equation, 1.5.7 Point of intersection of two lines, 1.5.8 Straight line models, 1.5.9 Intersection of a line and a curve, 1.5.10 Intersection of a line and a circle, 1.5.11 Equation of a circle, 1.5.12 Circle properties, 1.5.13 Parameter and parametric equations (A-level only), 1.5.14 Convert between cartesian and parametric forms (A-level only), 1.5.15 Equation of a circle in parametric form (A-level only), 1.5.16 Gradient of a parametric curve (A-level only), 1.5.17 Parametric equations in modelling (A-level only), and 1.5 Coordinate Geometry. Each one has a worked solution and a mark scheme showing where the marks go.