The path of a robotic arm in a two-dimensional manufacturing grid (X,Y)(X, Y)(X,Y) is defined by the parametric equations
X=2p−51−4p,Y=121−4p,p≠14 X = \frac{2p - 5}{1 - 4p}, \quad Y = \frac{12}{1 - 4p}, \quad p \neq \frac{1}{4} X=1−4p2p−5,Y=1−4p12,p=41where p p\,p is a control parameter.
Show that the robotic arm moves along a straight line.
The arm's path intersects a safety perimeter defined by the line Y=3X+10Y = 3X + 10Y=3X+10. Determine the XXX-coordinate of this point of intersection.
143 exam-style questions on OCR A Level Maths 1.3 Coordinate Geometry in the x-y Plane, covering 1.3.1 Straight lines, 1.3.2 Parallel and perpendicular lines, 1.3.3 Straight line models, 1.3.4 Coordinate geometry of a circle, 1.3.5 Centre and radius by completing the square, 1.3.6 Circle properties, 1.3.7 Parametric equations of curves (A-level only), 1.3.8 Parametric equations in context (A-level only), and 1.3 Coordinate Geometry in the x-y Plane. Each one has a worked solution and a mark scheme showing where the marks go.