Find the angle between the lines represented by the equation 2x^2 - 3xy + y^2 =

Find the angle between the lines represented by the equation 2x^2 - 3xy + y^2 = 0.

Find the angle between the lines represented by the equation 2x^2 - 3xy + y^2 = 0.

Step 1: Start with the given equation of the lines: 2x^2 - 3xy + y^2 = 0.
Step 2: Recognize that this is a quadratic equation in x and y, which represents two lines.
Step 3: To find the slopes of the lines, we need to factor the equation or use the quadratic formula.
Step 4: Rewrite the equation in the form of ax^2 + bxy + cy^2 = 0, where a = 2, b = -3, c = 1.
Step 5: Use the formula for the slopes of the lines: m1, m2 = [(-b ± √(b^2 - 4ac)) / (2a)].
Step 6: Calculate b^2 - 4ac: (-3)^2 - 4(2)(1) = 9 - 8 = 1.
Step 7: Now calculate the slopes: m1, m2 = [3 ± √1] / 4 = [3 ± 1] / 4.
Step 8: This gives us two slopes: m1 = (3 + 1) / 4 = 1 and m2 = (3 - 1) / 4 = 0.5.
Step 9: Now, use the formula for the angle between the lines: tan(θ) = |(m1 - m2) / (1 + m1*m2)|.
Step 10: Substitute m1 and m2 into the formula: tan(θ) = |(1 - 0.5) / (1 + 1*0.5)| = |0.5 / 1.5| = 1/3.
Step 11: Finally, to find the angle θ, take the arctan of 1/3.

Quadratic Equations – Understanding how to interpret and manipulate quadratic equations to find slopes of lines.
Slopes of Lines – Calculating the slopes (m1 and m2) from the given quadratic equation to determine the angle between the lines.
Angle Between Lines – Using the formula for the tangent of the angle between two lines based on their slopes.