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Edexcel A-Level Mathematics A2 P4.3 Coordinate Geometry in the X Y Plane Questions

Practise converting parametric curves into Cartesian form and using algebra to locate points, ranges and intersections.

Syllabus
First assessment 2019
Course
Mathematics YMA01
Level
A2

Exam points

  • eliminate t from parametric equations to form a Cartesian equation without trig terms
  • find intercepts or intersection points by substituting curve parameters into a line equation

Covered syllabus points

P4.3.1 - Parametric equations of curves and conversion between

Question 1

[Maximum number: 1]
Figure 3

Figure 3

Figure 3 shows a sketch of the curve C with parametric equations

x=6t−3sin⁡2ty=2cos⁡t0⩽t⩽π2x=6 t-3 \sin 2 t \quad y=2 \cos t \quad 0 \leqslant t \leqslant \frac{\pi}{2}

The curve meets the y-axis at 2 and the x-axis at k, where k is a constant.

State the value of k.

Question 2

[Maximum number: 2]
Figure 2

Figure 2

Figure 2 shows a sketch of the curve defined by the parametric equations

x=t2+2ty=2t(3−t)a⩽t⩽bx=t^{2}+2 t \quad y=\frac{2}{t(3-t)} \quad a \leqslant t \leqslant b

where a and b are constants.
The ends of the curve lie on the line with equation y=1

Find the value of a and the value of b

The region R, shown shaded in Figure 2, is bounded by the curve and the line with equation y=1

Question 3

[Maximum number: 5]
Figure 4

Figure 4

Figure 4 shows a sketch of the curve C with parametric equations

x=sec⁡ty=3tan⁡(t+π3)π6<t<π2x=\sec t \quad y=\sqrt{3} \tan \left(t+\frac{\pi}{3}\right) \quad \frac{\pi}{6}<t<\frac{\pi}{2}

Show that all points on C satisfy the equation

y=Ax2+B3x2−34−3x2y=\frac{A x^{2}+B \sqrt{3 x^{2}-3}}{4-3 x^{2}}

where A and B are constants to be found.

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