The angle between the lines represented by the equation 3x^2 - 4xy + 2y^2 = 0 is

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Question: The angle between the lines represented by the equation 3x^2 - 4xy + 2y^2 = 0 is:

Options:

Correct Answer: 60 degrees

Solution:

Using the formula tan(θ) = |(m1 - m2) / (1 + m1*m2)|, we find that the angle is 60 degrees.

The angle between the lines represented by the equation 3x^2 - 4xy + 2y^2 = 0 is:

The angle between the lines represented by the equation 3x^2 - 4xy + 2y^2 = 0 is:

Step 1: Identify the given equation, which is 3x^2 - 4xy + 2y^2 = 0. This is a quadratic equation in x and y.
Step 2: Recognize that this equation represents two lines. To find the slopes of these lines, we need to factor the equation.
Step 3: Factor the equation 3x^2 - 4xy + 2y^2 = 0. This can be factored as (3x + 2y)(x - y) = 0.
Step 4: From the factored form, identify the two lines: 3x + 2y = 0 and x - y = 0.
Step 5: Find the slopes (m1 and m2) of these lines. For the first line (3x + 2y = 0), rearranging gives y = -3/2 x, so m1 = -3/2. For the second line (x - y = 0), rearranging gives y = x, so m2 = 1.
Step 6: Use the formula for the angle between two lines: tan(θ) = |(m1 - m2) / (1 + m1*m2)|.
Step 7: Substitute m1 and m2 into the formula: tan(θ) = |(-3/2 - 1) / (1 + (-3/2)*1)|.
Step 8: Simplify the expression: tan(θ) = |(-5/2) / (-1/2)| = |5| = 5.
Step 9: Find θ by taking the arctan of 5. This gives θ ≈ 60 degrees.

Angle Between Lines – The question tests the understanding of how to find the angle between two lines represented by a quadratic equation.
Quadratic Equations – It assesses the ability to interpret and manipulate quadratic equations to extract information about their geometric properties.
Slope of Lines – The concept of slopes (m1 and m2) of the lines derived from the quadratic equation is crucial for applying the angle formula.