A robotic arm is programmed to sweep across a linear welding track. Its horizontal position xxx (in cm) relative to a central sensor is modeled by the equation
x=5tan(y−π4)x∈R,−π4<y<3π4 x = 5\tan\left(y - \frac{\pi}{4}\right) \quad x \in \mathbb{R}, \quad -\frac{\pi}{4} < y < \frac{3\pi}{4} x=5tan(y−4π)x∈R,−4π<y<43πwhere y y\,y is the angle of rotation of the arm in radians.
Show that
dydx=ax2+b \frac{dy}{dx} = \frac{a}{x^2 + b} dxdy=x2+bawhere a a\,a and b b\,b are integers to be found.
The point P P\,P on the curve C C\,C has yyy-coordinate π2\displaystyle \frac{\pi}{2}2π. The tangent to C C\,C at P P\,P crosses the xxx-axis at the point QQQ. Find the exact xxx-coordinate of QQQ.
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.