For the lines represented by the equation x^2 - 2xy + y^2 = 0, find the slopes o

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Question: For the lines represented by the equation x^2 - 2xy + y^2 = 0, find the slopes of the lines.

Options:

Correct Answer: 1, -1

Solution:

The slopes can be found by solving the quadratic equation formed by the coefficients of x^2, xy, and y^2.

For the lines represented by the equation x^2 - 2xy + y^2 = 0, find the slopes of the lines.

For the lines represented by the equation x^2 - 2xy + y^2 = 0, find the slopes of the lines.

Correct Answer: 1 and 1

Step 1: Start with the given equation: x^2 - 2xy + y^2 = 0.
Step 2: Rewrite the equation in a standard form for a quadratic equation in terms of x: x^2 - 2xy + y^2 = 0 can be seen as a quadratic in x.
Step 3: Identify the coefficients: Here, the coefficient of x^2 is 1, the coefficient of xy is -2y, and the coefficient of y^2 is y^2.
Step 4: Use the quadratic formula to find the values of x: The quadratic formula is x = (-b ± √(b² - 4ac)) / 2a.
Step 5: Substitute a = 1, b = -2y, and c = y^2 into the quadratic formula.
Step 6: Calculate the discriminant: b² - 4ac = (-2y)² - 4(1)(y^2) = 4y² - 4y² = 0.
Step 7: Since the discriminant is 0, there is one repeated solution for x, which means the lines are coincident.
Step 8: Solve for the slope: The slope of the line can be found from the equation y = mx + b, where m is the slope.
Step 9: From the original equation, we can factor it as (x - y)(x - y) = 0, which gives us the line y = x.
Step 10: The slope of the line y = x is 1.

Quadratic Equations – Understanding how to manipulate and solve quadratic equations, particularly in the context of conic sections.
Slopes of Lines – Finding the slopes of lines represented by a quadratic equation in two variables.
Factoring and Roots – Using factoring techniques to find the roots of the equation, which correspond to the slopes of the lines.