Let `0 lt p lt q` and `a ne 0` such that the equation `px^(2) +4 lambda xy +qy^(2) +4a (x+y +1) = 0` represents a pair of straight lines, then a can lie in the interval

Options

A. (a) `(-oo,oo)`
B. (b) `(-oo,p]`
C. (c) `[p,q]`
D. (d) `[q,oo)`

Explanation

To solve the problem, we need to analyze the given equation and determine the interval in which \( a \) lies, given that the equation represents a pair of straight lines.

### Step-by-Step Solution:

1. **Identify the coefficients**:
The given equation is:
\[ px^2 + 4\lambda xy + qy^2 + 4a(x + y + 1) = 0 \]
We can rewrite it in the form of the general conic equation:
\[ Ax^2 + 2Hxy + By^2 + 2Gx + 2Fy + C = 0 \]
where:
- \( A = p \)
- \( 2H = 4\lambda \) → \( H = 2\lambda \)
- \( B = q \)
- \( 2G = 4a \) → \( G = 2a \)
- \( 2F = 4a \) → \( F = 2a \)
- \( C = 4a \)

2. **Condition for a pair of straight lines**:
The equation represents a pair of straight lines if the following condition is satisfied:
\[ ABC + 2FHG - AF^2 - BG^2 - CH^2 = 0 \]

3. **Substituting the coefficients**:
Substitute the values of \( A, B, C, F, G, H \):
\[ p \cdot q \cdot (4a) + 2(2a)(2\lambda)(2a) - p(2a)^2 - q(2a)^2 - (4a)(2\lambda)^2 = 0 \]
Simplifying gives:
\[ 4apq + 8a^2\lambda - 4pa^2 - 4qa^2 - 16a\lambda^2 = 0 \]

4. **Rearranging the equation**:
Rearranging the equation:
\[ 4a(pq + 2a\lambda - pa - qa - 4\lambda^2) = 0 \]
Since \( a \neq 0 \), we can divide by \( 4a \):
\[ pq + 2a\lambda - pa - qa - 4\lambda^2 = 0 \]

5. **Forming a quadratic in \( a \)**:
Rearranging gives us:
\[ 2a\lambda - pa - qa = -pq + 4\lambda^2 \]
This can be rewritten as:
\[ a(2\lambda - p - q) = -pq + 4\lambda^2 \]
Thus:
\[ a = \frac{-pq + 4\lambda^2}{2\lambda - p - q} \]

6. **Finding the discriminant**:
For \( a \) to be real, the discriminant of the quadratic in \( a \) must be non-negative:
\[ D = (2\lambda - p - q)^2 - 4(-pq + 4\lambda^2) \geq 0 \]

7. **Analyzing the intervals**:
The critical points occur when \( a = p \) and \( a = q \). The expression \( (a - p)(a - q) \) will be positive when \( a < p \) or \( a > q \). Thus, the intervals for \( a \) are:
\[ (-\infty, p] \cup [q, \infty) \]

### Conclusion:
The value of \( a \) can lie in the intervals:
\[ a \in (-\infty, p] \cup [q, \infty) \]