Question

Problem: If one of the lines represented by \(ax^2 + 2hxy + by^2 = 0\) bisects the angle between the positive coordinate axes, then identify the correct relationship between \(a\), \(b\), and \(h\). Identify the correct option from the following:

• A. \(a + b = 2h\)
• B. \(a - b = 2h\)
• C. \(a + 2h + b = 0\)
• D. \(a + 2h - b = 0\)

Hint:

For a pair of lines \(ax^2 + 2hxy + by^2 = 0\), the lines are perpendicular if \(a + b = 0\).

Answer 1

The Correct Option is A

Step 1: Understand the problem
The equation \(ax^2 + 2hxy + by^2 = 0\) represents a pair of straight lines through the origin. The angle between the positive coordinate axes is \(90^\circ\), so the angle between the lines must also be \(90^\circ\). Step 2: Condition for perpendicular lines
For the lines given by \(ax^2 + 2hxy + by^2 = 0\) to be perpendicular, the condition is \(a + b = 0\). Here, \(a + b = 0 \implies b = -a\). Step 3: Determine the slopes and apply the condition
The equation can be written as \(a x^2 + 2h xy + (-a) y^2 = 0\). The slopes \(m_1\) and \(m_2\) satisfy \(m_1 m_2 = \frac{a}{-a} = -1\), confirming the lines are perpendicular. Now, check the sum of slopes: \(m_1 + m_2 = \frac{-2h}{-a} = \frac{2h}{a}\). However, we use the perpendicularity condition directly to test options. Step 4: Test the options
With \(a + b = 0\), and assuming \(h = 0\) (as derived), option (1) \(a + b = 2h \implies 0 = 2(0) \implies 0 = 0\), which holds true.