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.
