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 polar coordinates
In polar coordinates, we have:
- \( x = r \cos \theta \)
- \( y = r \sin \theta \)

### Step 2: Substitute into the equation
Now, substitute \( x \) and \( y \) into the given equation:
\[ x^2 + y^2 = ax \]
This becomes:
\[ (r \cos \theta)^2 + (r \sin \theta)^2 = a(r \cos \theta) \]

### Step 3: Simplify the left side
The left side simplifies as follows:
\[ (r \cos \theta)^2 + (r \sin \theta)^2 = r^2 \cos^2 \theta + r^2 \sin^2 \theta \]
Factoring out \( r^2 \):
\[ r^2 (\cos^2 \theta + \sin^2 \theta) \]

### Step 4: Use the Pythagorean identity
Using the identity \( \cos^2 \theta + \sin^2 \theta = 1 \), we have:
\[ r^2 \cdot 1 = r^2 \]
So the equation becomes:
\[ r^2 = a(r \cos \theta) \]

### Step 5: Rearranging the equation
We can rearrange this equation:
\[ r^2 = ar \cos \theta \]

### Step 6: Divide by \( r \) (assuming \( r \neq 0 \))
Dividing both sides by \( r \) gives us:
\[ r = a \cos \theta \]

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

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