Transform the equation r = 2 a cos `theta` to cartesian form.

Answer

To transform the equation \( r = 2a \cos \theta \) into Cartesian form, we can follow these steps:

### Step 1: Start with the given polar equation
The equation we have is:
\[ r = 2a \cos \theta \]

### Step 2: Multiply both sides by \( r \)
To eliminate \( r \) from the right side, we multiply both sides by \( r \):
\[ r^2 = 2ar \cos \theta \]

### Step 3: Substitute polar to Cartesian conversions
We know from polar coordinates that:
- \( r^2 = x^2 + y^2 \)
- \( r \cos \theta = x \)

Substituting these into our equation gives:
\[ x^2 + y^2 = 2ar \]

### Step 4: Substitute \( r \) in terms of \( x \) and \( y \)
Since \( r = \sqrt{x^2 + y^2} \), we can substitute this into the equation:
\[ x^2 + y^2 = 2a\sqrt{x^2 + y^2} \]

### Step 5: Square both sides to eliminate the square root
To remove the square root, we square both sides:
\[ (x^2 + y^2)^2 = (2a)^2(x^2 + y^2) \]

### Step 6: Simplify the equation
Expanding both sides gives:
\[ x^4 + 2x^2y^2 + y^4 = 4a^2(x^2 + y^2) \]

### Step 7: Rearranging the equation
Rearranging this, we get:
\[ x^4 + 2x^2y^2 + y^4 - 4a^2x^2 - 4a^2y^2 = 0 \]

This is the Cartesian form of the given polar equation.

### Final Result
Thus, the Cartesian form of the equation \( r = 2a \cos \theta \) is:
\[ x^4 + 2x^2y^2 + y^4 - 4a^2x^2 - 4a^2y^2 = 0 \]

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