If the lines represented by the equation 2x^2 + 3xy + y^2 = 0 are intersecting, what is the condition on the coefficients?

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Q: If the lines represented by the equation 2x^2 + 3xy + y^2 = 0 are intersecting, what is the condition on the coefficients?

Solution: The lines intersect if the discriminant D = b^2 - 4ac > 0.

Steps: 9

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: Rewrite the equation in the standard form of a quadratic equation: ax^2 + bxy + cy^2 = 0.
Step 4: Identify the coefficients: a = 2, b = 3, c = 1.
Step 5: Use the formula for the discriminant D = b^2 - 4ac.
Step 6: Substitute the values of a, b, and c into the discriminant formula: D = (3)^2 - 4(2)(1).
Step 7: Calculate D: D = 9 - 8 = 1.
Step 8: Determine the condition for the lines to intersect: the discriminant D must be greater than 0 (D > 0).
Step 9: Since D = 1, which is greater than 0, the lines intersect.