If the lines represented by the equation 6x^2 - 5xy + y^2 = 0 are intersecting,

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Question: If the lines represented by the equation 6x^2 - 5xy + y^2 = 0 are intersecting, what is the nature of the roots?

Options:

Correct Answer: Real and distinct

Solution:

The nature of the roots can be determined by the discriminant of the quadratic equation.

If the lines represented by the equation 6x^2 - 5xy + y^2 = 0 are intersecting, what is the nature of the roots?

If the lines represented by the equation 6x^2 - 5xy + y^2 = 0 are intersecting, what is the nature of the roots?

Correct Answer: Real and distinct roots

Step 1: Identify the given equation, which is 6x^2 - 5xy + y^2 = 0. This is a quadratic equation in terms of x and y.
Step 2: Rewrite the equation in standard quadratic form, which is Ax^2 + Bxy + Cy^2 = 0. Here, A = 6, B = -5, and C = 1.
Step 3: Calculate the discriminant (D) using the formula D = B^2 - 4AC.
Step 4: Substitute the values of A, B, and C into the discriminant formula: D = (-5)^2 - 4(6)(1).
Step 5: Simplify the calculation: D = 25 - 24 = 1.
Step 6: Analyze the value of the discriminant. Since D > 0, it indicates that the roots are real and distinct.
Step 7: Conclude that the nature of the roots is that they are real and distinct, which means the lines represented by the equation intersect.

Quadratic Equations – Understanding the nature of roots based on the discriminant of a quadratic equation.
Discriminant – The discriminant (D = b^2 - 4ac) determines whether the roots are real and distinct, real and equal, or complex.
Conic Sections – Recognizing that the given equation represents a pair of lines and analyzing their intersection.