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 condition 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. **Identify the Coefficients:**
The given equation is in the form of a conic section. We can identify the coefficients:
- Coefficient of \( x^2 \) is \( A = 1 \)
- Coefficient of \( y^2 \) is \( B = k_1 \)
- Coefficient of \( y \) is \( C = 2k_2 \)
- Constant term is \( D = -a^2 \)

2. **Condition for Perpendicular Lines:**
For the equation to represent a pair of perpendicular straight lines, the condition is given by:
\[ A + B = 0 \]
Substituting the values, we have:
\[ 1 + k_1 = 0 \]
Therefore, we can solve for \( k_1 \):
\[ k_1 = -1 \]

3. **Using the General Condition for Pair of Straight Lines:**
The equation represents a pair of straight lines if the following condition holds:
\[ ABC + 2FGH - AF^2 - BG^2 - CH^2 = 0 \]
Here, we need to identify \( A, B, C, F, G, H \):
- \( A = 1 \)
- \( B = k_1 = -1 \)
- \( C = -a^2 \)
- \( F = k_2 \)
- \( G = 0 \) (since there is no \( x \) term)
- \( H = 0 \) (since there is no \( y^2 \) term)

4. **Substituting the Values:**
Now substituting these values into the condition:
\[ (1)(-1)(-a^2) + 2(k_2)(0)(0) - (1)(k_2^2) - (-1)(0) - (-a^2)(0) = 0 \]
Simplifying this gives:
\[ a^2 - k_2^2 = 0 \]
Thus, we can solve for \( k_2 \):
\[ k_2^2 = a^2 \implies k_2 = \pm a \]

5. **Final Condition:**
Therefore, 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:
- \( k_1 = -1 \)
- \( k_2 = \pm a \)

### Summary:
The equation represents a pair of perpendicular straight lines if:
- \( k_1 = -1 \)
- \( k_2 = \pm a \)