If the lines represented by the equation 3x^2 + 2xy - y^2 = 0 intersect at an angle of 60 degrees, what is the value of the coefficient of xy?

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Q: If the lines represented by the equation 3x^2 + 2xy - y^2 = 0 intersect at an angle of 60 degrees, what is the value of the coefficient of xy?

Solution: Using the formula for the angle between two lines, we can derive the coefficient of xy that satisfies the angle condition.

Steps: 8

Step 1: Start with the given equation 3x^2 + 2xy - y^2 = 0. This represents two lines.
Step 2: Identify the general form of a conic section, which is Ax^2 + Bxy + Cy^2 = 0. Here, A = 3, B = 2, and C = -1.
Step 3: Use the formula for the angle (θ) between two lines given by the equation: tan(θ) = |(B)/(A - C)|.
Step 4: Substitute the values of A, B, and C into the formula: tan(60 degrees) = |(2)/(3 - (-1))|.
Step 5: Calculate tan(60 degrees), which is √3. So, we have √3 = |(2)/(4)|.
Step 6: Simplify the right side: |(2)/(4)| = 1/2. Now we have √3 = 1/2, which is not correct.
Step 7: To find the correct coefficient of xy (B), we need to set up the equation: √3 = |B/(3 - (-1))| = |B/4|.
Step 8: Solve for B: B = 4√3 or B = -4√3. Since we want the coefficient of xy, we can take B = 4√3.