The lines represented by the equation 4x^2 - 12xy + 9y^2 = 0 are:

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Question: The lines represented by the equation 4x^2 - 12xy + 9y^2 = 0 are:

Options:

Correct Answer: Perpendicular

Solution:

The lines are perpendicular if the product of their slopes is -1. We can find the slopes from the equation and check this condition.

The lines represented by the equation 4x^2 - 12xy + 9y^2 = 0 are:

The lines represented by the equation 4x^2 - 12xy + 9y^2 = 0 are:

Correct Answer: The lines are perpendicular.

Step 1: Start with the given equation: 4x^2 - 12xy + 9y^2 = 0.
Step 2: Recognize that this is a quadratic equation in terms of x and y, which can represent two lines.
Step 3: Rewrite the equation in the standard form of a conic section, which is Ax^2 + Bxy + Cy^2 = 0.
Step 4: Identify the coefficients: A = 4, B = -12, C = 9.
Step 5: Use the formula to find the slopes of the lines: m1, m2 = (B ± √(B^2 - 4AC)) / (2A).
Step 6: Calculate B^2 - 4AC: (-12)^2 - 4(4)(9) = 144 - 144 = 0.
Step 7: Since B^2 - 4AC = 0, this means the lines are coincident (the same line) and not distinct.
Step 8: Therefore, we cannot find two different slopes to check if they are perpendicular.

Quadratic Equations – Understanding how to analyze and factor quadratic equations to find their roots, which can represent lines.
Slope of Lines – Knowing how to derive the slopes of lines from their equations and the condition for perpendicularity.
Conic Sections – Recognizing that the given equation represents a pair of lines, which is a specific case of conic sections.