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

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

Options:

Correct Answer: Perpendicular

Solution:

Factoring gives (2x - 3y)(2x - 3y) = 0, indicating the lines are coincident.

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

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

Step 1: Start with the equation 4x^2 - 12xy + 9y^2 = 0.
Step 2: Recognize that this is a quadratic equation in terms of x and y.
Step 3: Try to factor the equation. Look for two expressions that multiply to give the original equation.
Step 4: Notice that 4x^2 can be written as (2x)(2x) and 9y^2 as (3y)(3y).
Step 5: Rewrite the equation as (2x - 3y)(2x - 3y) = 0.
Step 6: This shows that both factors are the same, meaning the lines are coincident (they overlap completely).

Quadratic Equations – Understanding how to factor quadratic equations to find the roots or lines represented by them.
Coincident Lines – Recognizing that when a quadratic factors into a perfect square, the lines represented are coincident.