The lines represented by the equation 2x^2 + 3xy + y^2 = 0 are:

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Question: The lines represented by the equation 2x^2 + 3xy + y^2 = 0 are:

Options:

Correct Answer: Intersecting

Solution:

To determine the nature of the lines, we can analyze the discriminant of the quadratic equation.

The lines represented by the equation 2x^2 + 3xy + y^2 = 0 are:

The lines represented by the equation 2x^2 + 3xy + y^2 = 0 are:

Correct Answer: Two intersecting lines

Step 1: Identify the given equation, which is 2x^2 + 3xy + y^2 = 0.
Step 2: Recognize that this is a quadratic equation in two variables (x and y).
Step 3: To analyze the nature of the lines, we need to find the discriminant of the equation.
Step 4: The discriminant (D) for a quadratic equation Ax^2 + Bxy + Cy^2 = 0 is calculated using the formula D = B^2 - 4AC.
Step 5: In our equation, A = 2, B = 3, and C = 1.
Step 6: Substitute the values into the discriminant formula: D = (3)^2 - 4(2)(1).
Step 7: Calculate D: D = 9 - 8 = 1.
Step 8: Analyze the value of D: Since D > 0, this means the equation represents two distinct lines.

Quadratic Equations – Understanding how to analyze quadratic equations in two variables, particularly in the context of conic sections.
Discriminant – Using the discriminant to determine the nature of the roots of a quadratic equation, which can indicate whether the lines are real, coincident, or complex.
Conic Sections – Recognizing that the given equation represents a conic section and understanding the implications of its coefficients.