Determine the condition for the lines represented by ax^2 + 2hxy + by^2 = 0 to b

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Question: Determine the condition for the lines represented by ax^2 + 2hxy + by^2 = 0 to be parallel.

Options:

Correct Answer: h^2 = ab

Solution:

The lines are parallel if the discriminant of the quadratic equation is zero, which leads to the condition h^2 = ab.

Determine the condition for the lines represented by ax^2 + 2hxy + by^2 = 0 to be parallel.

Determine the condition for the lines represented by ax^2 + 2hxy + by^2 = 0 to be parallel.

Step 1: Understand that the equation ax^2 + 2hxy + by^2 = 0 represents two lines.
Step 2: Recognize that for two lines to be parallel, they must not intersect.
Step 3: Recall that the condition for a quadratic equation to have repeated roots (which means the lines are parallel) is that the discriminant must be zero.
Step 4: The discriminant (D) for the equation ax^2 + 2hxy + by^2 = 0 is given by D = (2h)^2 - 4ab.
Step 5: Set the discriminant equal to zero for the lines to be parallel: (2h)^2 - 4ab = 0.
Step 6: Simplify the equation: 4h^2 - 4ab = 0.
Step 7: Divide the entire equation by 4: h^2 - ab = 0.
Step 8: Rearrange the equation to find the condition: h^2 = ab.

Quadratic Forms – Understanding how the equation ax^2 + 2hxy + by^2 = 0 represents a pair of lines and how their properties can be analyzed using discriminants.
Discriminant Condition – The condition for the lines to be parallel is derived from the discriminant of the quadratic equation being zero.