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

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

Options:

Correct Answer: Intersecting

Solution:

The lines intersect at the origin and are not parallel, hence they are intersecting.

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

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

Step 1: Start with the equation x^2 + 2xy + y^2 = 0.
Step 2: Recognize that this is a quadratic equation in two variables (x and y).
Step 3: Factor the equation. Notice that it can be rewritten as (x + y)(x + y) = 0.
Step 4: Set each factor equal to zero. This gives us x + y = 0.
Step 5: Solve for y in terms of x. From x + y = 0, we get y = -x.
Step 6: This means there is a line represented by y = -x.
Step 7: The line passes through the origin (0,0).
Step 8: Since the equation represents a single line (y = -x) that can be counted as two identical lines, they intersect at the origin.
Step 9: Conclude that the lines are not parallel because they are the same line.

Quadratic Equations – The equation x^2 + 2xy + y^2 = 0 represents a conic section, specifically a pair of lines.
Intersection of Lines – Understanding how to determine if lines intersect and their relationship (parallel, intersecting, or coincident).
Factoring Quadratics – The ability to factor the quadratic equation to find the lines it represents.