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

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

Options:

Correct Answer: Perpendicular

Solution:

The discriminant is negative, indicating that the lines are perpendicular.

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

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

Correct Answer: Perpendicular lines

Step 1: Identify the equation given, which is 5x^2 - 6xy + 5y^2 = 0.
Step 2: Recognize that this is a quadratic equation in two variables (x and y).
Step 3: To analyze the lines represented by this equation, we need to calculate the discriminant.
Step 4: The discriminant (D) for a quadratic equation Ax^2 + Bxy + Cy^2 = 0 is given by the formula D = B^2 - 4AC.
Step 5: In our equation, A = 5, B = -6, and C = 5.
Step 6: Substitute these values into the discriminant formula: D = (-6)^2 - 4(5)(5).
Step 7: Calculate (-6)^2, which is 36.
Step 8: Calculate 4(5)(5), which is 100.
Step 9: Now, subtract: D = 36 - 100 = -64.
Step 10: Since the discriminant is negative (D < 0), this indicates that the lines represented by the equation are perpendicular.

Quadratic Forms – The equation represents a conic section, specifically a pair of lines, which can be analyzed using the discriminant.
Discriminant Analysis – The discriminant of a quadratic equation helps determine the nature of the roots, which in this case indicates the relationship between the lines.
Perpendicular Lines – Understanding the conditions under which two lines are perpendicular based on their slopes derived from the quadratic equation.