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If the lines represented by the equation 3x^2 + 2xy - y^2 = 0 intersect at an an

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Question: 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?

Options:

  1. 2
  2. 3
  3. 4
  4. 5

Correct Answer: 2

Solution: 

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

If the lines represented by the equation 3x^2 + 2xy - y^2 = 0 intersect at an an

Practice Questions

Q1
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?
  1. 2
  2. 3
  3. 4
  4. 5

Questions & Step-by-Step Solutions

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?
  • 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.
  • Angle Between Lines – Understanding how to calculate the angle between two lines represented by a quadratic equation.
  • Quadratic Equations – Recognizing the form of a quadratic equation and how it relates to the intersection of lines.
  • Coefficients in Quadratics – Identifying the significance of coefficients in determining the properties of the lines represented by the equation.
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