Question

If the equation \[ ax^2 + 2hxy + by^2 + 2gx + 2fy = 0 \] has one line as the bisector of the angle between the co-ordinate axes, then \[ (a + b)^2 = 4(h^2 + g^2) \]

• A. \( (a + b)^2 = 4(h^2 + g^2) \)
• B. \( (a + b)^2 = 4h^2 \)
• C. \( (a + b)^2 = 4(h^2 + f^2) \)
• D. \( (a + b)^2 = 4(h^2 + g^2 + f^2) \)

Hint:

For conic equations involving angle bisectors, use the condition \( (a + b)^2 = 4(h^2 + g^2) \) to identify the relationship between the coefficients.

Answer 1

The Correct Option is A

Step 1: Understand the condition for the bisector.
For the equation of the conic, the condition for one of the lines being the bisector of the angle between the axes is given by \( (a + b)^2 = 4(h^2 + g^2) \). This can be derived by considering the condition for the angle between two lines and equating it to the angle between the coordinate axes.

Step 2: Conclusion.
Thus, the correct answer is \( (a + b)^2 = 4(h^2 + g^2) \), corresponding to option (A).