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The pair of lines represented by the equation x^2 - 4xy + 3y^2 = 0 are:

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Question: The pair of lines represented by the equation x^2 - 4xy + 3y^2 = 0 are:

Options:

  1. Parallel
  2. Perpendicular
  3. Intersecting
  4. Coincident

Correct Answer: Perpendicular

Solution: 

To determine the nature of the lines, we can find the slopes from the equation. The product of the slopes will help us conclude if they are perpendicular.

The pair of lines represented by the equation x^2 - 4xy + 3y^2 = 0 are:

Practice Questions

Q1
The pair of lines represented by the equation x^2 - 4xy + 3y^2 = 0 are:
  1. Parallel
  2. Perpendicular
  3. Intersecting
  4. Coincident

Questions & Step-by-Step Solutions

The pair of lines represented by the equation x^2 - 4xy + 3y^2 = 0 are:
  • Step 1: Start with the given equation: x^2 - 4xy + 3y^2 = 0.
  • Step 2: This equation is a quadratic in terms of x and y, which can be factored to find the slopes of the lines.
  • Step 3: Rewrite the equation in the standard form: (x - ay)(x - by) = 0, where a and b are the slopes.
  • Step 4: To factor, we need to find two numbers that multiply to 3 (the coefficient of y^2) and add to -4 (the coefficient of xy).
  • Step 5: The numbers that work are -3 and -1, so we can factor the equation as (x - 3y)(x - y) = 0.
  • Step 6: From the factored form, we can see the slopes of the lines are 3 and 1 (from x = 3y and x = y).
  • Step 7: To check if the lines are perpendicular, calculate the product of the slopes: 3 * 1 = 3.
  • Step 8: Since the product of the slopes is not -1, the lines are not perpendicular.
  • Quadratic Equations – Understanding how to analyze and factor quadratic equations to find the slopes of the lines they represent.
  • Slope of Lines – Calculating the slopes from the factored form of the quadratic equation to determine the relationship between the lines.
  • Perpendicular Lines – Recognizing that the product of the slopes of two lines can indicate if they are perpendicular.
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