The transform equation of `r^(2)cos^(2)theta = a^(2)cos 2theta` to cartesian form is`(x^(2)+y^(2))x^(2)=a^(2)lambda`, then value of `lambda` is

Options

A. `y^(2)-x^(2)`
B. `x^(2)-y^(2)`
C. xy
D. `x^(2)y^(2)`

Answer

To convert the polar equation \( r^2 \cos^2 \theta = a^2 \cos 2\theta \) into Cartesian form and find the value of \( \lambda \), we can follow these steps:

### Step 1: Rewrite the polar equation
The given polar equation is:
\[ r^2 \cos^2 \theta = a^2 \cos 2\theta \]
We know that \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \). Thus, we can rewrite the equation as:
\[ r^2 \cos^2 \theta = a^2 (\cos^2 \theta - \sin^2 \theta) \]

### Step 2: Substitute \( r \) in terms of \( x \) and \( y \)
In Cartesian coordinates, we have:
\[ x = r \cos \theta \quad \text{and} \quad y = r \sin \theta \]
From this, we can express \( r^2 \) as:
\[ r^2 = x^2 + y^2 \]
Now substitute \( r^2 \) into the equation:
\[ (x^2 + y^2) \cos^2 \theta = a^2 (\cos^2 \theta - \sin^2 \theta) \]

### Step 3: Express \( \cos^2 \theta \) and \( \sin^2 \theta \) in terms of \( x \) and \( y \)
Using the relationships:
\[ \cos^2 \theta = \frac{x^2}{r^2} = \frac{x^2}{x^2 + y^2} \]
\[ \sin^2 \theta = \frac{y^2}{r^2} = \frac{y^2}{x^2 + y^2} \]
Substituting these into the equation gives:
\[ (x^2 + y^2) \left(\frac{x^2}{x^2 + y^2}\right) = a^2 \left(\frac{x^2}{x^2 + y^2} - \frac{y^2}{x^2 + y^2}\right) \]

### Step 4: Simplify the equation
The left side simplifies to:
\[ x^2 \]
The right side simplifies to:
\[ a^2 \left(\frac{x^2 - y^2}{x^2 + y^2}\right) \]
Thus, we have:
\[ x^2 = a^2 \frac{x^2 - y^2}{x^2 + y^2} \]

### Step 5: Multiply both sides by \( x^2 + y^2 \)
To eliminate the fraction, multiply both sides by \( x^2 + y^2 \):
\[ x^2 (x^2 + y^2) = a^2 (x^2 - y^2) \]

### Step 6: Rearranging the equation
This can be rearranged to:
\[ x^2 + y^2 = a^2 \frac{x^2 - y^2}{x^2} \]
From the original problem statement, we can see that this matches the form:
\[ (x^2 + y^2)x^2 = a^2 \lambda \]
Thus, we can identify:
\[ \lambda = x^2 - y^2 \]

### Final Answer
The value of \( \lambda \) is:
\[ \lambda = x^2 - y^2 \]