The lines represented by the equation 2x^2 + 3xy + y^2 = 0 are:
Correct Answer: Two intersecting lines
- Step 1: Identify the given equation, which is 2x^2 + 3xy + y^2 = 0.
- Step 2: Recognize that this is a quadratic equation in two variables (x and y).
- Step 3: To analyze the nature of the lines, we need to find the discriminant of the equation.
- Step 4: The discriminant (D) for a quadratic equation Ax^2 + Bxy + Cy^2 = 0 is calculated using the formula D = B^2 - 4AC.
- Step 5: In our equation, A = 2, B = 3, and C = 1.
- Step 6: Substitute the values into the discriminant formula: D = (3)^2 - 4(2)(1).
- Step 7: Calculate D: D = 9 - 8 = 1.
- Step 8: Analyze the value of D: Since D > 0, this means the equation represents two distinct lines.
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