The angle between the lines represented by the equation 2x^2 + 3xy + y^2 = 0 is:

The angle between the lines represented by the equation 2x^2 + 3xy + y^2 = 0 is:

Step 1: Identify the given equation, which is 2x^2 + 3xy + y^2 = 0. This is a quadratic equation in two variables (x and y).
Step 2: Recognize that this equation represents two lines. We can find the angle between these lines using a specific formula.
Step 3: The formula to find the angle (θ) between two lines represented by the equation Ax^2 + Bxy + Cy^2 = 0 is given by: tan(θ) = |(B) / (A - C)|.
Step 4: In our equation, A = 2, B = 3, and C = 1. Substitute these values into the formula: tan(θ) = |3 / (2 - 1)|.
Step 5: Calculate the value: tan(θ) = |3 / 1| = 3.
Step 6: Now, find the angle θ using the arctan function: θ = arctan(3).
Step 7: Use a calculator or trigonometric tables to find θ. This gives us approximately 71.57 degrees.
Step 8: Since the angle between two lines can be acute or obtuse, we can also find the supplementary angle. The acute angle is 60 degrees.

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