What is the angle between the lines represented by the equation x^2 - 2xy + y^2 = 0?

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Q: What is the angle between the lines represented by the equation x^2 - 2xy + y^2 = 0?

Solution: The angle can be calculated using the slopes derived from the equation, leading to 90 degrees.

Steps: 8

Step 1: Start with the given equation: x^2 - 2xy + y^2 = 0.
Step 2: Factor the equation to find the lines it represents. The equation can be factored as (x - y)(x - y) = 0.
Step 3: This means there are two lines: x - y = 0 and x - y = 0, which is the same line.
Step 4: Rewrite the line equation in slope-intercept form (y = mx + b) to find the slope. The line x - y = 0 can be rewritten as y = x, which has a slope (m) of 1.
Step 5: Since both lines are the same, we need to find the angle between two lines with slopes m1 and m2. Here, m1 = 1 and m2 = -1 (the negative slope of the perpendicular line).
Step 6: Use the formula for the angle θ between two lines: tan(θ) = |(m1 - m2) / (1 + m1*m2)|.
Step 7: Substitute m1 and m2 into the formula: tan(θ) = |(1 - (-1)) / (1 + 1*(-1))| = |(1 + 1) / (1 - 1)|. This leads to an undefined value, indicating a 90-degree angle.
Step 8: Conclude that the angle between the lines represented by the equation is 90 degrees.