2- Find the Polar equation for the following curves: (i) $$ 9 x^{2}+9 y^{2}=36 \quad(2, \pi) $$ (ii) $$ \left(x^{2}+y^{2}\right)^{3}=4 x^{2} y^{2} $$ (iii) $$ x^{2}+y^{2}=2 a y $$ (iv) $$ x^{2}-y^{2}=a^{2} $$

Question

UpStudy AI · Accepted Answer

To convert the given Cartesian equations into polar equations, we will use the relationships $$ x = r \cos \theta $$ and $$ y = r \sin \theta $$, where $$ r $$ is the radial distance from the origin and $$ \theta $$ is the angle measured from the positive x-axis.

### (i) $$ 9x^2 + 9y^2 = 36 $$

First, we can simplify the equation:

$$
x^2 + y^2 = 4
$$

Now, substituting $$ x $$ and $$ y $$ with their polar equivalents:

$$
r^2 = 4
$$

Taking the square root gives:

$$
r = 2
$$

Thus, the polar equation is:

$$
r = 2
$$

### (ii) $$ (x^2 + y^2)^3 = 4x^2y^2 $$

Substituting $$ x^2 + y^2 = r^2 $$ and $$ x^2y^2 = (r \cos \theta)(r \sin \theta) = r^2 \cos^2 \theta \sin^2 \theta $$:

$$
(r^2)^3 = 4(r^2 \cos^2 \theta \sin^2 \theta)
$$

This simplifies to:

$$
r^6 = 4r^2 \cos^2 \theta \sin^2 \theta
$$

Dividing both sides by $$ r^2 $$ (assuming $$ r \neq 0 $$):

$$
r^4 = 4 \cos^2 \theta \sin^2 \theta
$$

Using the identity $$ \sin^2 \theta = \frac{1 - \cos(2\theta)}{2} $$ and $$ \cos^2 \theta = \frac{1 + \cos(2\theta)}{2} $$:

$$
r^4 = 4 \left(\frac{1 - \cos(2\theta)}{2}\right)\left(\frac{1 + \cos(2\theta)}{2}\right)
$$

This simplifies to:

$$
r^4 = \frac{4}{4} (1 - \cos^2(2\theta)) = \sin^2(2\theta)
$$

Thus, the polar equation is:

$$
r^4 = \sin^2(2\theta)
$$

### (iii) $$ x^2 + y^2 = 2ay $$

Substituting $$ x^2 + y^2 = r^2 $$ and $$ y = r \sin \theta $$:

$$
r^2 = 2a(r \sin \theta)
$$

Dividing both sides by $$ r $$ (assuming $$ r \neq 0 $$):

$$
r = 2a \sin \theta
$$

Thus, the polar equation is:

$$
r = 2a \sin \theta
$$

### (iv) $$ x^2 - y^2 = a^2 $$

Substituting $$ x^2 = r^2 \cos^2 \theta $$ and $$ y^2 = r^2 \sin^2 \theta $$:

$$
r^2 \cos^2 \theta - r^2 \sin^2 \theta = a^2
$$

Factoring out $$ r^2 $$:

$$
r^2 (\cos^2 \theta - \sin^2 \theta) = a^2
$$

Using the identity $$ \cos^2 \theta - \sin^2 \theta = \cos(2\theta) $$:

$$
r^2 \cos(2\theta) = a^2
$$

Thus, the polar equation is:

$$
r^2 = \frac{a^2}{\cos(2\theta)}
$$

### Summary of Polar Equations

1. $$ r = 2 $$
2. $$ r^4 = \sin^2(2\theta) $$
3. $$ r = 2a \sin \theta $$
4. $$ r^2 = \frac{a^2}{\cos(2\theta)} $$
1. $$ r = 2 $$ 2. $$ r^4 = \sin^2(2\theta) $$ 3. $$ r = 2a \sin \theta $$ 4. $$ r^2 = \frac{a^2}{\cos(2\theta)} $$

2- Find the Polar equation for the following curves:
(i)
(ii)
(iii)
(iv)

Upstudy AI Solution

Answer

Solution

Beyond the Answer

Related Questions

Latest Pre Calculus Questions

2- Find the Polar equation for the following curves: (i) (ii) (iii) (iv)

Upstudy AI Solution

Answer

Solution

Beyond the Answer

Related Questions

Latest Pre Calculus Questions

2- Find the Polar equation for the following curves:
(i)
(ii)
(iii)
(iv)