Edexcel A-Level Mathematics A2 Fp2 7 1 Polar Coordinates and Polar Curves Questions

Practise reading and sketching polar curves, linking r and θ values to points, lines and parameters in bounded regions.

Exam points

substitute a given θ into a polar curve and line equation to determine an exact parameter

[Maximum number: 2]

Figure 1

Figure 1 shows a sketch of the curve C with polar equation

r=1+\cos\theta,\qquad 0\le\theta\le\pi

and the line l with polar equation

r=k\sec\theta,\qquad 0\le\theta<\frac{\pi}{2}

where k is a positive constant.

Given that
- C and l intersect at the point P
- $OP=1+\frac{\sqrt3}{2}$

determine the exact value of k.

The finite region R, shown shaded in Figure 1, is bounded by C, the initial line and l.

At P,

1+\cos\theta=1+\frac{\sqrt3}{2}

so $\cos\theta=\frac{\sqrt3}{2}$ . Since $r=k\sec\theta$ ,

1+\frac{\sqrt3}{2}=k\cdot\frac{2}{\sqrt3}

Hence

k=\frac{\sqrt3}{2}+\frac34

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