The lines represented by the equation x^2 - 6xy + 9y^2 = 0 are:

The lines represented by the equation x^2 - 6xy + 9y^2 = 0 are:

Step 1: Start with the equation x^2 - 6xy + 9y^2 = 0.
Step 2: Recognize that this is a quadratic equation in terms of x and y.
Step 3: Try to factor the equation. Look for two identical factors.
Step 4: Notice that (x - 3y)(x - 3y) = (x - 3y)^2.
Step 5: Rewrite the equation as (x - 3y)^2 = 0.
Step 6: Understand that (x - 3y)^2 = 0 means x - 3y = 0.
Step 7: Solve x - 3y = 0 to find the line represented by the equation.
Step 8: The line can be written as x = 3y, which means it is a single line.
Step 9: Since both factors are the same, the lines are coincident (the same line).

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