What is the angle between the lines represented by the equation 2x^2 + 3xy - 2y^

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

Options:

Correct Answer: 90 degrees

Solution:

Using the formula for the angle between two lines, we find that the angle is 90 degrees.

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

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

Correct Answer: 90 degrees

Step 1: Identify the given equation, which is 2x^2 + 3xy - 2y^2 = 0. This is a quadratic equation in x and y.
Step 2: Recognize that this equation represents two lines. We can factor it to find the equations of the lines.
Step 3: Factor the equation. We can rewrite it as (2x - y)(x + 2y) = 0. This gives us two lines: 2x - y = 0 and x + 2y = 0.
Step 4: Find the slopes of the two lines. For the first line (2x - y = 0), rearranging gives y = 2x, so the slope (m1) is 2. For the second line (x + 2y = 0), rearranging gives y = -0.5x, so the slope (m2) is -0.5.
Step 5: Use the formula for the angle (θ) between two lines: tan(θ) = |(m1 - m2) / (1 + m1*m2)|. Plug in the slopes: tan(θ) = |(2 - (-0.5)) / (1 + 2*(-0.5))|.
Step 6: Calculate the values: tan(θ) = |(2 + 0.5) / (1 - 1)| = |2.5 / 0|. Since the denominator is 0, this means the lines are perpendicular.
Step 7: Conclude that if the lines are perpendicular, the angle between them is 90 degrees.

Quadratic Equations – Understanding how to interpret and manipulate quadratic equations to find the angle between lines.
Angle Between Lines – Applying the formula for the angle between two lines derived from their slopes.
Factoring and Roots – Factoring the quadratic equation to find the slopes of the lines represented.