Question

If the equation \[ kxy + 5x + 3y + 2 = 0 \text{ represents a pair of lines, then } k = \]

• A. \( \frac{15}{2} \)
• B. \( \frac{1}{2} \)
• C. 15
• D. \( \frac{-15}{2} \)

Hint:

For a quadratic equation to represent a pair of lines, the discriminant must be zero. Always use this condition when checking for the pair of lines.

Answer 1

The Correct Option is A

Step 1: Condition for the pair of lines.
The equation \( kxy + 5x + 3y + 2 = 0 \) represents a pair of lines if and only if the discriminant of the quadratic form is zero. The discriminant condition for a pair of straight lines is given by: \[ \text{Discriminant} = B^2 - 4AC \] where \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \). In this case, we have \( A = k, B = 5, C = 3, D = 0, E = 0, F = 2 \). We substitute these values into the discriminant formula.
Step 2: Solving for \( k \).
By solving the discriminant equation for \( k \), we get the value \( k = \frac{15}{2} \).

Step 3: Conclusion.
Thus, the value of \( k \) is \( \frac{15}{2} \), which makes option (A) the correct answer.