`x ^(2) (lamda ^(2) - 4 lamda + 3) +y ^(2) (lamda ^(2) - 6 lamda +5) =1` will represent an ellipse if `lamda` lies in the interval

Options

A. `(-oo, 1) uu (5, oo)`
B. `(-oo,-1) (5,oo)`
C. R
D. none of these

Answer

To determine the values of \( \lambda \) for which the given equation represents an ellipse, we start with the equation:

\[ x^2 (\lambda^2 - 4\lambda + 3) + y^2 (\lambda^2 - 6\lambda + 5) = 1 \]

### Step 1: Identify the conditions for an ellipse

For the equation to represent an ellipse, the coefficients of \( x^2 \) and \( y^2 \) must be positive. Thus, we need:

1. \( \lambda^2 - 4\lambda + 3 > 0 \)
2. \( \lambda^2 - 6\lambda + 5 > 0 \)

### Step 2: Solve the first inequality

We start with the first inequality:

\[ \lambda^2 - 4\lambda + 3 > 0 \]

Factoring gives:

\[ (\lambda - 1)(\lambda - 3) > 0 \]

To find the intervals, we analyze the sign changes around the roots \( \lambda = 1 \) and \( \lambda = 3 \):

- For \( \lambda < 1 \): both factors are negative, so the product is positive.
- For \( 1 < \lambda < 3 \): one factor is negative and the other is positive, so the product is negative.
- For \( \lambda > 3 \): both factors are positive, so the product is positive.

Thus, the solution for the first inequality is:

\[ \lambda \in (-\infty, 1) \cup (3, \infty) \]

### Step 3: Solve the second inequality

Next, we solve the second inequality:

\[ \lambda^2 - 6\lambda + 5 > 0 \]

Factoring gives:

\[ (\lambda - 1)(\lambda - 5) > 0 \]

Analyzing the sign changes around the roots \( \lambda = 1 \) and \( \lambda = 5 \):

- For \( \lambda < 1 \): both factors are negative, so the product is positive.
- For \( 1 < \lambda < 5 \): one factor is negative and the other is positive, so the product is negative.
- For \( \lambda > 5 \): both factors are positive, so the product is positive.

Thus, the solution for the second inequality is:

\[ \lambda \in (-\infty, 1) \cup (5, \infty) \]

### Step 4: Find the intersection of the two intervals

Now, we find the intersection of the two intervals obtained from the inequalities:

1. From the first inequality: \( (-\infty, 1) \cup (3, \infty) \)
2. From the second inequality: \( (-\infty, 1) \cup (5, \infty) \)

The intersection is:

\[ \lambda \in (-\infty, 1) \cup (5, \infty) \]

### Conclusion

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

\[ \lambda \in (-\infty, 1) \cup (5, \infty) \]