The pair of lines represented by the equation 5x^2 + 6xy + 5y^2 = 0 are:

₹0.0

Question: The pair of lines represented by the equation 5x^2 + 6xy + 5y^2 = 0 are:

Options:

Correct Answer: Imaginary

Solution:

The discriminant of the quadratic equation is negative, indicating imaginary lines.

The pair of lines represented by the equation 5x^2 + 6xy + 5y^2 = 0 are:

The pair of lines represented by the equation 5x^2 + 6xy + 5y^2 = 0 are:

Step 1: Identify the equation given, which is 5x^2 + 6xy + 5y^2 = 0.
Step 2: Recognize that this is a quadratic equation in two variables (x and y).
Step 3: To analyze the lines represented by this equation, we need to calculate the discriminant.
Step 4: The discriminant (D) for a quadratic equation of the form Ax^2 + Bxy + Cy^2 = 0 is given by the formula D = B^2 - 4AC.
Step 5: In our equation, A = 5, B = 6, and C = 5.
Step 6: Substitute these values into the discriminant formula: D = 6^2 - 4(5)(5).
Step 7: Calculate 6^2, which is 36.
Step 8: Calculate 4(5)(5), which is 100.
Step 9: Now, subtract: D = 36 - 100 = -64.
Step 10: Since the discriminant is negative (D < 0), this indicates that the lines represented by the equation are imaginary.

Quadratic Equations – Understanding the nature of roots of quadratic equations, particularly in the context of conic sections.
Discriminant – The discriminant (D = b^2 - 4ac) determines the nature of the roots of a quadratic equation, where a positive D indicates real and distinct roots, D = 0 indicates real and repeated roots, and negative D indicates complex roots.
Conic Sections – The study of the geometric properties of conic sections, including lines, circles, ellipses, parabolas, and hyperbolas.