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

Answer

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

### Step 1: Write down the polar to Cartesian conversions
In polar coordinates, we have:
- \( x = r \cos \theta \)
- \( y = r \sin \theta \)
- \( r = \sqrt{x^2 + y^2} \)

### Step 2: Substitute \( r \) and \( \cos \theta \) in the given equation
The given equation is:
\[ r = 2a \cos \theta \]
We can express \( \cos \theta \) in terms of \( x \) and \( r \):
\[ \cos \theta = \frac{x}{r} \]
Substituting this into the equation gives:
\[ r = 2a \left(\frac{x}{r}\right) \]

### Step 3: Multiply both sides by \( r \)
To eliminate the fraction, we multiply both sides by \( r \):
\[ r^2 = 2a x \]

### Step 4: Replace \( r^2 \) with \( x^2 + y^2 \)
Now we can replace \( r^2 \) with \( x^2 + y^2 \):
\[ x^2 + y^2 = 2a x \]

### Step 5: Rearrange the equation
Rearranging the equation gives us:
\[ x^2 + y^2 - 2ax = 0 \]

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

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