# Polar curves

A **polar curve** is a shape constructed using the polar coordinate system. Polar curves are defined by points that are a varaible distance from the origin (the pole) depending on the angle measured off the positive \(x\)-axis. Polar curves can describe familiar Cartesian shapes such as ellipses as well as some unfamiliar shapes such as cardioids and lemniscates.

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## Polar Coordinates System

Each point in the polar coordinate system is given by \( (r, \theta ) \), where \( r \) is the radius (distance from the pole to the point) and \( \theta \) is the angle that is made between the radius and the x-axis.

To convert back and forth between polar and rectangular coordinates, we have the following formulas:

\[\begin{align} x &= r \cos \theta \\ y &= r \sin \theta\\ x^2 + y^2 &= r^2 \\ \tan \theta &= \dfrac{y}{x}.\ _\square \end{align}\]

Convert \((1, 2)\) into polar coordinates.

Convert \( \left( 4, \dfrac{\pi}{4} \right) \) into rectangular coordinates.

## Polar Equations

The equation of a curve is \(x^2+y^2-x=\sqrt{x^2+y^2}\).

\((\text{i})\) Find the polar equation of the curve in the form \(r=f(\theta)\).

\((\text{ii})\) Sketch the curve.

\((\text{iii})\) The line \(x+2y=2\) divides the region enclosed by the curve into two parts. Find the ratio of the two areas. (If the ratio is \(a:b\) in its simplest form, input \(a+b\) as your answer.)

There are **3** marks available for part (i), **2** marks for part (ii) and **6** marks for part (iii). In total, this question is worth **15.3%** of all available marks in the paper.

This is part of the set OCR A Level Problems.

## Types of Polar Curves

- Cardioid
- Limacon
- Rose petal curves
- Archimedes spiral
- Lemniscate

## Derivatives of Polar Curves

To find \( \dfrac{dy}{dx} \) given that \(x = r \cos \theta \) and \( y = r \sin \theta \), we can find \( \dfrac{dy}{d\theta} \) and \( \dfrac{dx}{d\theta } \). Upon dividing the former by the latter, we get \( \dfrac{dy}{dx} \).

For a polar curve,

\[ \dfrac{dy}{dx} = \dfrac{\dfrac{dr}{d\theta} \sin \theta + r \cos \theta}{\dfrac{dr}{d\theta} \cos \theta - r\sin \theta}.\ _\square \]

**Important facts to note:**

- When \( \dfrac{dy}{d\theta} = 0\), we have horizontal tangent lines.
- When \( \dfrac{dx}{d\theta}= 0 \), we have vertical tangent lines.
- When you find the tangent lines at the pole, let's say the slope to the tangent is \( m \). Then the equation of that tangent line will be \( \theta = \arctan m. \)

Given \( r = 1 + \cos \theta \), find the equation of all tangent lines at the pole.

## Integrals of Polar Curves

(I will try to put in a graphic later)

(add explanation) \[ \displaystyle \dfrac{1}{2} \int_{\alpha}^{\beta} r^{2} \, d\theta \]

- Area between two polar curves

## Arc Length of Polar Curves

For a polar curve \( r = f(\theta ) \), given that the polar curve's first derivative is everywhere continuous, and the domain does not cause the polar curve to retrace itself, the arc length on \( \alpha \leqslant \theta \leqslant \beta \) is given by

\[ \displaystyle \int_{\alpha}^{\beta} \sqrt{r^2 + \left(\dfrac{dr}{d\theta}\right)^2} \, d\theta.\ _\square \]