The lines represented by the equation 5x^2 - 6xy + 5y^2 = 0 intersect at:

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Question: The lines represented by the equation 5x^2 - 6xy + 5y^2 = 0 intersect at:

Options:

Correct Answer: (0,0)

Solution:

The lines intersect at the origin (0,0) as derived from the equation.

The lines represented by the equation 5x^2 - 6xy + 5y^2 = 0 intersect at:

The lines represented by the equation 5x^2 - 6xy + 5y^2 = 0 intersect at:

Step 1: Start with the equation 5x^2 - 6xy + 5y^2 = 0.
Step 2: Recognize that this is a quadratic equation in terms of x and y.
Step 3: Factor the equation to find the lines it represents.
Step 4: The equation can be factored as (5x - 5y)(x - y) = 0.
Step 5: Set each factor equal to zero: 5x - 5y = 0 and x - y = 0.
Step 6: Solve the first equation: 5x = 5y, which simplifies to x = y.
Step 7: Solve the second equation: x = y.
Step 8: Both equations give the same line, indicating they intersect along this line.
Step 9: To find the intersection point, substitute y = 0 into x = y, giving x = 0.
Step 10: Therefore, the lines intersect at the origin (0,0).

Quadratic Equations – The question involves analyzing a quadratic equation in two variables, which can represent conic sections or lines.
Intersection of Lines – Understanding how to find the points where lines represented by a quadratic equation intersect.