The equation `(x^(2))/(10-lambda)+(y^(2))/(6-lambda)=1` represents

Options

A. a hyperbola if `lambdalt6`
B. an ellipse if `lambdagt6`
C. a hyperbola if `6ltlambdalt10`
D. an ellipse if `0ltlambdalt6`

Answer

To determine the nature of the conic section represented by the equation

\[ \frac{x^2}{10 - \lambda} + \frac{y^2}{6 - \lambda} = 1, \]

we need to analyze the conditions under which this equation represents a hyperbola or an ellipse.

### Step 1: Identify the form of the equation
The given equation can be compared to the standard forms of conic sections:
- For a hyperbola: \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\)
- For an ellipse: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\)

In our case, we have:

\[ \frac{x^2}{10 - \lambda} + \frac{y^2}{6 - \lambda} = 1. \]

This indicates that both denominators must be positive for the equation to represent an ellipse.

### Step 2: Conditions for an ellipse
For the equation to represent an ellipse, both denominators must be positive:

1. \(10 - \lambda > 0 \Rightarrow \lambda < 10\)
2. \(6 - \lambda > 0 \Rightarrow \lambda < 6\)

Thus, for the equation to represent an ellipse, we need:

\[ \lambda < 6. \]

### Step 3: Conditions for a hyperbola
For the equation to represent a hyperbola, one of the denominators must be negative. This means that one of the conditions must be violated:

1. \(10 - \lambda > 0 \Rightarrow \lambda < 10\) (must remain positive)
2. \(6 - \lambda < 0 \Rightarrow \lambda > 6\) (must become negative)

Thus, for the equation to represent a hyperbola, we need:

\[ 6 < \lambda < 10. \]

### Step 4: Summary of conditions
- The equation represents an **ellipse** if \( \lambda < 6 \).
- The equation represents a **hyperbola** if \( 6 < \lambda < 10 \).

### Conclusion
Based on the analysis, we can conclude:
- The equation represents a hyperbola when \( 6 < \lambda < 10 \).
- The equation represents an ellipse when \( \lambda < 6 \).

### Final Answer
The correct options are:
- Hyperbola if \( 6 < \lambda < 10 \) (Option C).
- Ellipse if \( \lambda < 6 \) (Option D).

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