Determine the nature of the lines represented by the equation 7x^2 + 2xy + 3y^2

Determine the nature of the lines represented by the equation 7x^2 + 2xy + 3y^2 = 0.

Determine the nature of the lines represented by the equation 7x^2 + 2xy + 3y^2 = 0.

Step 1: Identify the equation given, which is 7x^2 + 2xy + 3y^2 = 0.
Step 2: Recognize that this is a quadratic equation in two variables (x and y).
Step 3: To determine the nature of the lines, we need to calculate the discriminant (D) of the equation.
Step 4: The discriminant for a general conic section Ax^2 + Bxy + Cy^2 = 0 is given by D = B^2 - 4AC.
Step 5: In our equation, A = 7, B = 2, and C = 3.
Step 6: Substitute these values into the discriminant formula: D = (2)^2 - 4(7)(3).
Step 7: Calculate D: D = 4 - 84 = -80.
Step 8: Analyze the value of D: Since D < 0, this indicates that the conic does not represent real lines.
Step 9: Conclude that the equation represents two complex lines that do not intersect in the real plane.

Conic Sections – Understanding the classification of conic sections based on their equations, particularly the nature of the lines represented by quadratic equations.
Discriminant Analysis – Using the discriminant of a quadratic equation to determine the nature of the roots, which in this case relates to the intersection of lines.