Transform the equation `x^(2)+y^(2)=ax` into polar form.

Answer

To transform the equation \( x^2 + y^2 = ax \) into polar form, we will follow these steps:

### Step 1: Substitute Cartesian coordinates with Polar coordinates
In polar coordinates, we have:
- \( x = r \cos \theta \)
- \( y = r \sin \theta \)

### Step 2: Substitute \( x \) and \( y \) in the equation
Substituting these into the given equation:
\[ x^2 + y^2 = ax \implies (r \cos \theta)^2 + (r \sin \theta)^2 = a(r \cos \theta) \]

### Step 3: Expand the left side
Expanding the left side:
\[ (r \cos \theta)^2 + (r \sin \theta)^2 = r^2 \cos^2 \theta + r^2 \sin^2 \theta \]

### Step 4: Factor out \( r^2 \)
Factoring out \( r^2 \):
\[ r^2 (\cos^2 \theta + \sin^2 \theta) \]

### Step 5: Use the Pythagorean identity
Using the Pythagorean identity \( \cos^2 \theta + \sin^2 \theta = 1 \):
\[ r^2 \cdot 1 = r^2 \]

### Step 6: Rewrite the equation
Now, the equation becomes:
\[ r^2 = a(r \cos \theta) \]

### Step 7: Rearrange the equation
Rearranging gives us:
\[ r^2 - a r \cos \theta = 0 \]

### Step 8: Factor out \( r \)
Factoring out \( r \):
\[ r(r - a \cos \theta) = 0 \]

### Step 9: Identify solutions
This gives us two solutions:
1. \( r = 0 \) (which corresponds to the origin)
2. \( r = a \cos \theta \) (which is the polar form of the equation)

### Final Polar Form
Thus, the polar form of the equation \( x^2 + y^2 = ax \) is:
\[ r = a \cos \theta \]

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