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

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`

Answer

To determine the values of \( k_1 \) and \( k_2 \) such that 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 General Form**: The given equation is \( x^2 + k_1 y^2 + 2k_2 y - a^2 = 0 \). This can be compared to the general conic section equation \( Ax^2 + By^2 + 2Gxy + 2Fx + 2Hy + C = 0 \).

2. **Extract Coefficients**: From the equation, we can identify:
- \( A = 1 \)
- \( B = k_1 \)
- \( C = -a^2 \)
- \( G = 0 \) (since there is no \( xy \) term)
- \( F = 0 \) (since there is no \( x \) term)
- \( H = k_2 \)

3. **Condition for Perpendicular Lines**: For the conic to represent a pair of perpendicular lines, the condition is:
\[ AB - G^2 = 0 \]
Substituting the values we have:
\[ 1 \cdot k_1 - 0^2 = 0 \implies k_1 = 0 \]

4. **Condition for Discriminant**: The discriminant condition for the conic section to represent real and distinct lines is given by:
\[ D = A \cdot C + 2H \cdot G \cdot F - A \cdot F^2 - B \cdot G^2 - C \cdot H^2 = 0 \]
Substituting the known values:
\[ 1 \cdot (-a^2) + 2 \cdot k_2 \cdot 0 \cdot 0 - 1 \cdot 0^2 - k_1 \cdot 0^2 - (-a^2) \cdot k_2^2 = 0 \]
Simplifying this gives:
\[ -a^2 - k_2^2 = 0 \implies k_2^2 = a^2 \]

5. **Finding \( k_2 \)**: From \( k_2^2 = a^2 \), we can conclude:
\[ k_2 = \pm a \]

6. **Final Values**: Therefore, we have:
- \( k_1 = -1 \)
- \( k_2 = \pm a \)

### Summary of Results:
- The values of \( k_1 \) and \( k_2 \) are:
- \( k_1 = -1 \)
- \( k_2 = a \) or \( k_2 = -a \)