Determine the condition for the lines represented by the equation 4x^2 + 4xy + y

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Question: Determine the condition for the lines represented by the equation 4x^2 + 4xy + y^2 = 0 to be coincident.

Options:

Correct Answer: b^2 - 4ac = 0

Solution:

For the lines to be coincident, the discriminant must be zero, i.e., b^2 - 4ac = 0.

Determine the condition for the lines represented by the equation 4x^2 + 4xy + y^2 = 0 to be coincident.

Determine the condition for the lines represented by the equation 4x^2 + 4xy + y^2 = 0 to be coincident.

Correct Answer: b^2 - 4ac = 0

Step 1: Start with the given equation: 4x^2 + 4xy + y^2 = 0.
Step 2: Identify the coefficients in the standard form of a quadratic equation Ax^2 + Bxy + Cy^2 = 0. Here, A = 4, B = 4, and C = 1.
Step 3: Write down the formula for the discriminant: D = B^2 - 4AC.
Step 4: Substitute the values of A, B, and C into the discriminant formula: D = (4)^2 - 4(4)(1).
Step 5: Calculate the discriminant: D = 16 - 16 = 0.
Step 6: Since the discriminant is zero, this means the lines represented by the equation are coincident.

Quadratic Equations – Understanding the representation of lines in a quadratic form and the conditions for their intersection.
Discriminant – Using the discriminant (b^2 - 4ac) to determine the nature of the roots of a quadratic equation.
Coincident Lines – Recognizing the condition for two lines to be coincident, which is when they overlap completely.