The pair of lines represented by the equation 2x^2 + 3xy + y^2 = 0 has:

The pair of lines represented by the equation 2x^2 + 3xy + y^2 = 0 has:

Step 1: Identify the equation given, which is 2x^2 + 3xy + y^2 = 0.
Step 2: Recognize that this is a quadratic equation in terms of x and y.
Step 3: To find the nature of the lines represented by this equation, we need to calculate the discriminant.
Step 4: The discriminant (D) for a quadratic equation Ax^2 + Bxy + Cy^2 = 0 is given by the formula D = B^2 - 4AC.
Step 5: In our equation, A = 2, B = 3, and C = 1.
Step 6: Substitute these values into the discriminant formula: D = (3)^2 - 4(2)(1).
Step 7: Calculate D: D = 9 - 8 = 1.
Step 8: Since the discriminant D is positive (1 > 0), this indicates that there are two distinct real roots.
Step 9: Therefore, the pair of lines represented by the equation is two distinct lines.

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