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1.5 Coordinate Geometry

1.5 Coordinate Geometry

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Question 62

A robotic welding head follows a path C C\,C in the xyxyxy-plane described by the parametric equations

x=2sin⁡θ+5cos⁡θ,y=4cos⁡2θ+2sin⁡θ,0≤θ≤π x = 2\sin \theta + 5\cos \theta, \quad y = 4\cos^2 \theta + 2\sin \theta, \quad 0 \le \theta \le \pi x=2sinθ+5cosθ,y=4cos2θ+2sinθ,0≤θ≤π
a.

Show that dydx=1\displaystyle \frac{dy}{dx} = 1dxdy​=1 where θ=0\theta = 0θ=0.

[3]
b.

The point P P\,P lies on C C\,C where θ=0\theta = 0θ=0.

Find the equation of the tangent to the path C C\,C at the point PPP, giving your answer in the form y=mx+cy = mx + cy=mx+c.

[3]
c.

The tangent to the path at P P\,P intersects the curve C C\,C again at the point QQQ.

Show that the value of θ \theta\,θ at point Q Q\,Q satisfies the equation

4cos⁡2θ−5cos⁡θ+1=0 4\cos^2 \theta - 5\cos \theta + 1 = 0 4cos2θ−5cosθ+1=0
[3]
d.

Hence find the exact value of the yyy-coordinate of QQQ.

[3]
Markscheme

1.5 Coordinate Geometry Questions

  1. A Level
  2. /Maths
  3. /1.5 Coordinate Geometry

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.

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