Find the condition for the lines represented by the equation 2x^2 + 3xy + y^2 = 0 to be parallel.

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Q: Find the condition for the lines represented by the equation 2x^2 + 3xy + y^2 = 0 to be parallel.

Solution: For the lines to be parallel, the discriminant D must be equal to 0.

Steps: 9

Step 1: Understand that the equation 2x^2 + 3xy + y^2 = 0 represents a pair of lines.
Step 2: Identify that for two lines to be parallel, they must have the same slope.
Step 3: Recall that the condition for a quadratic equation to represent two parallel lines is that the discriminant (D) must be equal to 0.
Step 4: The discriminant D for the general quadratic equation Ax^2 + Bxy + Cy^2 = 0 is given by 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 D is not equal to 0, the lines are not parallel under this condition.
Step 9: To find the condition for the lines to be parallel, set D = 0 and solve for the coefficients.