Q 10. - Calculus

Question

For each type of polar curve, give a general form of the polar equation for that curve:

(a) a line passing through the pole

(b) a circle centered at the pole

(c) a cardioid

(d) a limac¸on

(e) a rose with an odd number of petals

(f) a rose with an even number of petals

(g) a lemniscate

(h) the spiral of Archimedes

Step-by-Step Solution

Answer

Part (a) $r=\theta$

Part (b) $r=a\cos \theta$ or $r=a\sin \theta$

Part (c) $r=a\pm b\cos \theta ,r=a\pm b\sin \theta$ when $\left|\frac{a}{b}\right|=1$

Part (d) $r=a\pm b\cos \theta ,r=a\pm b\sin \theta$ when $\left|\frac{a}{b}\right|\ne 1$

Part (e) $r=a\sin k\theta ,r=a\cos k\theta$ where $k$ is odd.

Part (f) $r=a\sin k\theta ,r=a\cos k\theta$ where $k$ is even.

Part (a) Step 1: Given information

The goal is to write the general form of the polar equation.

Part (a) Step 1: Explanation

Consider different types of polar curves.

For each type of polar curve, the goal is to write the general form of the polar equation.

A line passing through the pole,

$r=\theta$ is the generic version of the polar equation for a line traveling through the pole.

Where $\theta$ is the angle at the pole or the origin.

Therefore the answer is $r=\theta$

Part (b) Step 1: Explanation

A circle centered at the pole,

A polar equation of the circle centered at the pole has the generic form $r=a\cos \theta$ or $r=a\sin \theta$. where $a\in \mathrm{ℝ}$

Therefore, the answer is $r=a\cos \theta$ or $r=a\sin \theta$

Part (c) Step 1: Explanation

A cardioid, 

The general form of a polar equation of the cardioid is,

$r=a\pm b\cos \theta ,r=a\pm b\sin \theta$ when $a,b\in \mathrm{ℝ}\left|\frac{a}{b}\right|=1$ and $a\ne 0,b\ne 0$

Therefore, the answer is $r=a\pm b\cos \theta ,r=a\pm b\sin \theta$ when $\left|\frac{a}{b}\right|=1$

Part (d) Step 1: Explanation

 A limacon,

The general form of a limacon polar equation is,

$r=a\pm b\cos \theta ,r=a\pm b\sin \theta$ when $a,b\in \mathrm{ℝ}\left|\frac{a}{b}\right|\ne 1$ and $a\ne 0,b\ne 0.$

Therefore, the answer is $r=a\pm b\cos \theta ,r=a\pm b\sin \theta$ when $\left|\frac{a}{b}\right|\ne 1$

Part (e) Step 1: Explanation

A rose with an unusually large number of petals,

A polar equation for a rose with an odd number of petals takes the following general form: $r=a\sin k\theta ,r=a\cos k\theta$ where $k$ is odd.

Therefore, the answer is $r=a\sin k\theta ,r=a\cos k\theta$ where $k$ is odd

Part (f) Step 1: Explanation

A rose with an even number of petals is called a petal rose.

The general form of a limacon polar equation is,

$r=a\sin k\theta ,r=a\cos k\theta$ where $k$ is even

Therefore, the answer is $r=a\sin k\theta ,r=a\cos k\theta$ where $k$is even .