For the lines represented by the equation 3x^2 - 2xy + y^2 = 0 to be parallel, t

For the lines represented by the equation 3x^2 - 2xy + y^2 = 0 to be parallel, the condition is:

For the lines represented by the equation 3x^2 - 2xy + y^2 = 0 to be parallel, the condition is:

Step 1: Understand that the equation 3x^2 - 2xy + y^2 = 0 represents a conic section, which can include lines.
Step 2: Recognize that for the lines to be parallel, the determinant of the coefficients of the quadratic terms must equal zero.
Step 3: Identify the coefficients from the equation: A = 3, B = -2, C = 1.
Step 4: Write the determinant formula for the coefficients: D = B^2 - 4AC.
Step 5: Substitute the values into the determinant formula: D = (-2)^2 - 4(3)(1).
Step 6: Calculate the determinant: D = 4 - 12 = -8.
Step 7: Since the determinant is not zero, the lines are not parallel. For them to be parallel, set D = 0 and solve for the condition.

Quadratic Equations – Understanding how to analyze and manipulate quadratic equations to determine the nature of their roots.
Parallel Lines – Recognizing the condition for lines to be parallel, which involves the relationship between their slopes.
Determinants – Using determinants to find conditions for the existence of parallel lines in a system of equations.