The curve C C\,C has parametric equations
x=2tx = 2tx=2t, y=t2−ky = t^2 - ky=t2−k
where k k\,k is a positive constant.
C C\,C meets the line y=xy = xy=x at two points, A A\,A and BBB.
Show that the xxx-coordinates of A A\,A and B B\,B satisfy
x2−4x−4k=0x^2 - 4x - 4k = 0x2−4x−4k=0
Given that the length of AB AB\,AB is 50\sqrt{50}50, find the value of kkk.
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.