Determine the condition for the lines represented by ax^2 + 2hxy + by^2 = 0 to b
Practice Questions
Q1
Determine the condition for the lines represented by ax^2 + 2hxy + by^2 = 0 to be perpendicular.
h^2 = ab
h^2 = -ab
a + b = 0
a - b = 0
Questions & Step-by-Step Solutions
Determine the condition for the lines represented by ax^2 + 2hxy + by^2 = 0 to be perpendicular.
Correct Answer: 2h = a + b, h^2 = -ab
Step 1: Understand that the equation ax^2 + 2hxy + by^2 = 0 represents two lines in the xy-plane.
Step 2: Recall that two lines are perpendicular if the product of their slopes is -1.
Step 3: For the given equation, the condition for the lines to be perpendicular can be derived from the coefficients a, b, and h.
Step 4: The condition for the lines to be perpendicular is given by the equation 2h = a + b.
Step 5: Rearranging this condition leads to h^2 = -ab, which is another way to express the perpendicularity condition.
Condition for Perpendicular Lines – The condition for two lines represented by a quadratic equation to be perpendicular involves the relationship between the coefficients of the equation.
Quadratic Forms – Understanding how the coefficients in a quadratic equation relate to the geometric properties of the lines it represents.
