If the lines represented by the equation 6x^2 - 5xy + y^2 = 0 are real and distinct, what is the condition on the coefficients?

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Q: If the lines represented by the equation 6x^2 - 5xy + y^2 = 0 are real and distinct, what is the condition on the coefficients?

Solution: The condition for the lines to be real and distinct is that the discriminant D must be greater than 0.

Steps: 8

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: Recognize that this equation can represent two lines if it can be factored into two linear equations.
Step 3: Understand that for the lines to be real and distinct, we need to check the discriminant (D) of the quadratic equation.
Step 4: The discriminant D for a quadratic equation Ax^2 + Bxy + Cy^2 = 0 is given by the formula D = B^2 - 4AC.
Step 5: In our equation, A = 6, B = -5, and C = 1. Substitute these values into the discriminant formula: D = (-5)^2 - 4(6)(1).
Step 6: Calculate D: D = 25 - 24 = 1.
Step 7: Since D = 1, which is greater than 0, the lines represented by the equation are real and distinct.
Step 8: Conclude that the condition for the lines to be real and distinct is that the discriminant D must be greater than 0.