For the lines represented by the equation 4x^2 - 4xy + y^2 = 0, the angle between them is:

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Q: For the lines represented by the equation 4x^2 - 4xy + y^2 = 0, the angle between them is:

Solution: The lines are at an angle of 45 degrees as the determinant of the coefficients gives a non-zero value.

Steps: 10

Step 1: Identify the given equation, which is 4x^2 - 4xy + y^2 = 0.
Step 2: Recognize that this equation represents two lines in the form Ax^2 + Bxy + Cy^2 = 0.
Step 3: Identify the coefficients: A = 4, B = -4, C = 1.
Step 4: Calculate the determinant using the formula D = B^2 - 4AC.
Step 5: Substitute the values into the determinant formula: D = (-4)^2 - 4(4)(1).
Step 6: Simplify the calculation: D = 16 - 16 = 0.
Step 7: Since the determinant is zero, it indicates that the lines are coincident or parallel, not intersecting at an angle.
Step 8: To find the angle between the lines, use the formula tan(θ) = |(m1 - m2) / (1 + m1*m2)|, where m1 and m2 are the slopes of the lines.
Step 9: Find the slopes of the lines from the equation, which can be derived from the quadratic formula.
Step 10: Calculate the angle θ using the slopes found in the previous step.