If the equation `(x^(2))/( 3 - lambda) + (y^(2))/(lambda - 8) + 1 = 0 ` represents an ellipse, then
(i) `lambda lt 8`
(ii) `lambda gt 3`
(iii) ` 3 lt lambda lt 8`
(iv) `lambda lt 3" or "lambda gt 8 `

Options

A. `lambda lt 8`
B. `lambda gt 3`
C. ` 3 lt lambda lt 8`
D. `lambda lt 3" or "lambda gt 8 `

Answer

To determine the values of \(\lambda\) for which the equation

\[ \frac{x^2}{3 - \lambda} + \frac{y^2}{\lambda - 8} + 1 = 0 \]

represents an ellipse, we will follow these steps:

### Step 1: Rearranging the Equation
First, we rearrange the given equation to isolate the terms involving \(x^2\) and \(y^2\):

\[ \frac{x^2}{3 - \lambda} + \frac{y^2}{\lambda - 8} = -1 \]

### Step 2: Multiplying by -1
Next, we multiply the entire equation by -1 to make the right-hand side positive:

\[ -\frac{x^2}{3 - \lambda} - \frac{y^2}{\lambda - 8} = 1 \]

This can be rewritten as:

\[ \frac{x^2}{\lambda - 3} + \frac{y^2}{8 - \lambda} = 1 \]

### Step 3: Identifying Conditions for an Ellipse
For the equation to represent an ellipse, both denominators must be positive:

1. **Condition 1**: \(\lambda - 3 > 0\)
- This implies:
\[ \lambda > 3 \]

2. **Condition 2**: \(8 - \lambda > 0\)
- This implies:
\[ \lambda < 8 \]

### Step 4: Combining the Conditions
Combining both conditions, we have:

\[ 3 < \lambda < 8 \]

### Conclusion
Thus, the values of \(\lambda\) for which the given equation represents an ellipse are:

\[ \lambda \in (3, 8) \]

The correct option is (iii) \(3 < \lambda < 8\).

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