Question

Find the polar equation of the line L perpendicular to the line angle=pi/3 at P(4,pi/3)

Answer 1

Answer on Question #51517 – Math – Analytic Geometry

Find the polar equation of the line L perpendicular to the line angle=pi/3 at P(4,pi/3)

Solution

Fig.1

Take the equation of a line in the normal form

$$x \cos \alpha + y \sin \alpha - p = 0$$

where $\alpha = \pi / 3$ and $p = 4$.

Transformation formulas are given by

$$\left\{ \begin{array}{l} x = r \cos \varphi \\ y = r \sin \varphi \end{array} \right.$$

Then

$$r \cos \varphi \cdot \cos \frac{\pi}{3} + r \sin \varphi \cdot \sin \frac{\pi}{3} - 4 = 0 \Rightarrow r = \frac{4}{\cos \left(\varphi - \frac{\pi}{3}\right)}$$

Answer: $r = \frac{4}{\cos \left(\varphi - \frac{\pi}{3}\right)}$

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