If `(lambda-2)x^(2)+4y^(2)=4` represents a rectangular hyperbola then `lambda=`

Options

A. 0
B. 1
C. `-2`
D. 3

Answer

To find the value of \( \lambda \) such that the equation \( (\lambda - 2)x^2 + 4y^2 = 4 \) represents a rectangular hyperbola, we can follow these steps:

### Step 1: Rewrite the given equation
Start by rewriting the equation in a standard form. The given equation is:
\[ (\lambda - 2)x^2 + 4y^2 = 4 \]
We can divide the entire equation by 4 to simplify it:
\[ \frac{(\lambda - 2)x^2}{4} + y^2 = 1 \]

### Step 2: Identify the coefficients
From the equation \( \frac{(\lambda - 2)x^2}{4} + y^2 = 1 \), we can identify the coefficients:
- Coefficient of \( x^2 \): \( \frac{\lambda - 2}{4} \)
- Coefficient of \( y^2 \): \( 1 \)

### Step 3: Set up the condition for a rectangular hyperbola
For a hyperbola to be rectangular, the sum of the coefficients of \( x^2 \) and \( y^2 \) must equal zero. Thus, we set up the equation:
\[ \frac{\lambda - 2}{4} + 1 = 0 \]

### Step 4: Solve for \( \lambda \)
Now, we solve the equation:
\[ \frac{\lambda - 2}{4} + 1 = 0 \]
Multiply through by 4 to eliminate the fraction:
\[ \lambda - 2 + 4 = 0 \]
Simplifying this gives:
\[ \lambda + 2 = 0 \]
Thus, we find:
\[ \lambda = -2 \]

### Conclusion
The value of \( \lambda \) such that the given equation represents a rectangular hyperbola is:
\[ \lambda = -2 \]

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