What is the condition for the lines represented by the equation 5x^2 + 4xy + 3y^2 = 0 to be parallel?

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Q: What is the condition for the lines represented by the equation 5x^2 + 4xy + 3y^2 = 0 to be parallel?

Solution: The condition for parallel lines is that the determinant of the coefficients must be zero.

Steps: 8

Step 1: Identify the given equation, which is 5x^2 + 4xy + 3y^2 = 0. This is a quadratic equation in two variables (x and y).
Step 2: Recognize that this equation can represent two lines if it can be factored into linear terms.
Step 3: The general form of a quadratic equation representing two lines is Ax^2 + Bxy + Cy^2 = 0, where A, B, and C are coefficients.
Step 4: In our equation, A = 5, B = 4, and C = 3.
Step 5: To find the condition for the lines to be parallel, we need to calculate the determinant of the coefficients, which is given by the formula: D = B^2 - 4AC.
Step 6: Substitute the values of A, B, and C into the determinant formula: D = (4)^2 - 4(5)(3).
Step 7: Calculate D: D = 16 - 60 = -44.
Step 8: Since the determinant D is not equal to zero, the lines are not parallel. For the lines to be parallel, we need D to be equal to zero.