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1.5 Coordinate Geometry

1.5 Coordinate Geometry

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Question 74

A high-precision industrial laser tracks a path C C\,C on a component, defined by the parametric equations

x=3cos⁡2t−6sin⁡t,y=6sin⁡t+5cos⁡t,0≤t≤2π x = 3\cos^2 t - 6\sin t, \quad y = 6\sin t + 5\cos t, \quad 0 \le t \le 2\pi x=3cos2t−6sint,y=6sint+5cost,0≤t≤2π
a.

Show that dydx=−1\displaystyle \frac{dy}{dx} = -1dxdy​=−1 at the point where t=πt = \pit=π.

[2]
b.

The point P P\,P lies on the path where t=πt = \pit=π.

Find the equation of the tangent to the laser's path at P P\,P in the form y=mx+cy = mx + cy=mx+c, where m m\,m and c c\,c are constants to be determined.

[3]
c.

The laser's path is such that the tangent at P P\,P intersects the path C C\,C again at the point QQQ.

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

3cos⁡2t+5cos⁡t+2=0 3\cos^2 t + 5\cos t + 2 = 0 3cos2t+5cost+2=0
[3]
d.

Hence find the exact possible values of the yyy-coordinate of point QQQ.

[2]
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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