The following steps are helpful in tracing a polar curve **r = ƒ(θ)**.

**1. Symmetry**

**(i) **The curve is symmetrical about the initial line if its equation remains unchanged when **θ** is replaced by **– θ**.

**(ii)** The curve is symmetrical about the line through the pole perpendicular to the initial line if its equation remains unchanged when **θ** is replaced by **π – θ**.

**(iii) **The curve is symmetrical about the pole if its equation remains unchanged when **θ** is replaced by **π + θ**.

**2. Pole**

The curves passes through the pole if **r = 0** for a value of **θ**.

**3. Solving the equation**

Solve the equation of **r **in terms of **θ** and see how **r** varies as **θ** varies. Make a table of corresponding values of **θ** and **r**. Many polar equations involve periodic functions such as **sin θ**, **cos θ** etc. In such cases we consider values of **θ** from **0** to **2π** alone.

**4. Asymptotes**

Find out the asymptotes of the curve, if any.

**5. Region**

Find out the region on the plane in which no part of the curve lies.

**Remark: **Sometimes a polar curve easily traced by converting it into a Cartesian curve by means of the transformation **x = r cos θ** and **y = r sin θ**.

**For example: **Consider a polar curve **r cos θ = a cos 2θ**.

On multiplying both sides by **r**^{2}, we get

**r cos θ. r ^{2} = ar^{2} (cos^{2} θ – sin^{2} θ)**

** x (x ^{2} + y^{2}) = a (x^{2} – y^{2}).**

which can be easily traced.

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