Transform to Cartesian coordinates the equations: `r^2=a^2cos2theta`

Answer

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

### Step 1: Recall the relationships between polar and Cartesian coordinates
In polar coordinates, we have:
- \( x = r \cos \theta \)
- \( y = r \sin \theta \)
- \( r^2 = x^2 + y^2 \)

### Step 2: Substitute \( \cos 2\theta \)
We know that:
\[ \cos 2\theta = \cos^2 \theta - \sin^2 \theta \]
Using the relationships from Step 1, we can express \( \cos^2 \theta \) and \( \sin^2 \theta \) in terms of \( x \) and \( y \):
\[ \cos^2 \theta = \left(\frac{x}{r}\right)^2 = \frac{x^2}{r^2} \]
\[ \sin^2 \theta = \left(\frac{y}{r}\right)^2 = \frac{y^2}{r^2} \]
Thus, we can write:
\[ \cos 2\theta = \frac{x^2}{r^2} - \frac{y^2}{r^2} = \frac{x^2 - y^2}{r^2} \]

### Step 3: Substitute \( \cos 2\theta \) into the original equation
Now, substituting this expression for \( \cos 2\theta \) into the original equation \( r^2 = a^2 \cos 2\theta \):
\[ r^2 = a^2 \left( \frac{x^2 - y^2}{r^2} \right) \]

### Step 4: Multiply both sides by \( r^2 \)
To eliminate \( r^2 \) from the denominator, multiply both sides by \( r^2 \):
\[ r^4 = a^2 (x^2 - y^2) \]

### Step 5: Substitute \( r^2 \) with \( x^2 + y^2 \)
Now, substitute \( r^2 \) with \( x^2 + y^2 \):
\[ (x^2 + y^2)^2 = a^2 (x^2 - y^2) \]

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