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.
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