Homogeneous Quadratic Equations

The general form of a homogeneous quadratic equation in and is: This equation represents a pair of lines passing through the origin if and only if the discriminant . To see why, we attempt to factorize the quadratic equation.

Step 1: Factorize the quadratic equation.
The quadratic equation can be factored as: Expanding: For this to match the given equation , we must have , which simplifies to: Thus, the condition ensures that the quadratic equation represents a pair of lines.

Step 2: Prove that does not represent a pair of lines.
The equation is a degenerate case where both and are non-negative, and their sum can only be zero if and . This does not represent two distinct lines, so it does not satisfy the condition for a pair of lines.

Final Answer: The equation represents a pair of lines passing through the origin if and only if . The equation does not represent a pair of lines.