Equation `x^(2) +k_(1)y^(2) +2k_(2)y = a^(2)` represents a pair of perpendicular straight lines if

Options

A. `k_(1) = 1, k_(2) = a`
B. `k_(1) = 1, k_(2) =- a`
C. `k_(1) = -1, k_(2) =- a`
D. `k_(1) =- 1, k_(2) = a`

Explanation

To determine the conditions under which the equation \( x^2 + k_1 y^2 + 2k_2 y = a^2 \) represents a pair of perpendicular straight lines, we can follow these steps:

### Step-by-Step Solution

1. **Rearranging the Equation**:
Start with the given equation:
\[ x^2 + k_1 y^2 + 2k_2 y - a^2 = 0 \]
This is a general form of the combined equation of two straight lines.

2. **Identifying Coefficients**:
Compare the given equation with the general form of the combined equation of straight lines:
\[ Ax^2 + 2Hxy + By^2 + 2Gx + 2Fy + C = 0 \]
From this comparison, we identify:
- \( A = 1 \)
- \( B = k_1 \)
- \( C = -a^2 \)
- \( H = 0 \)
- \( G = 0 \)
- \( F = k_2 \)

3. **Condition for Perpendicular Lines**:
For the lines to be perpendicular, the condition is:
\[ A + B = 0 \]
Substituting the values we found:
\[ 1 + k_1 = 0 \]
This simplifies to:
\[ k_1 = -1 \]

4. **Using the Condition for Pair of Straight Lines**:
The general condition for the equation to represent a pair of straight lines is:
\[ ABC + 2FGH - A F^2 - B G^2 - C H^2 = 0 \]
Substituting the identified coefficients:
\[ 1 \cdot k_1 \cdot (-a^2) + 2 \cdot k_2 \cdot 0 \cdot 0 - 1 \cdot k_2^2 - k_1 \cdot 0^2 - (-a^2) \cdot 0^2 = 0 \]
This simplifies to:
\[ -k_1 a^2 - k_2^2 = 0 \]

5. **Substituting \( k_1 \)**:
Now substituting \( k_1 = -1 \):
\[ -(-1) a^2 - k_2^2 = 0 \]
This simplifies to:
\[ a^2 - k_2^2 = 0 \]
Thus:
\[ k_2^2 = a^2 \]
Taking the square root gives:
\[ k_2 = a \quad \text{or} \quad k_2 = -a \]

### Final Result
The conditions for the equation \( x^2 + k_1 y^2 + 2k_2 y = a^2 \) to represent a pair of perpendicular straight lines are:
\[ k_1 = -1 \quad \text{and} \quad k_2 = a \text{ or } k_2 = -a \]