Question

The equation of the pair of angle bisectors of the line \( x^2 - 4xy - 5y^2 = 0 \) is:

• A. \( x^2 - 3xy + y^2 = 0 \)
• B. \( x^2 + 4xy - y^2 = 0 \)
• C. \( x^2 + 3xy - y^2 = 0 \)
• D. \( x^2 - 3xy - y^2 = 0 \)

Hint:

Use the identity for angle bisectors of conic pairs: eliminate constant to get new homogeneous equation.

Answer 1

The Correct Option is C

Given: \( x^2 - 4xy - 5y^2 = 0 \) is a homogeneous equation representing pair of straight lines.
The angle bisectors of the pair of lines represented by \( ax^2 + 2hxy + by^2 = 0 \) are given by:
\[ \frac{x^2 - y^2}{a - b} = \frac{2xy}{2h} \Rightarrow (x^2 - y^2)(2h) = 2xy(a - b) \]
Here: \( a = 1 \), \( 2h = -4 \Rightarrow h = -2 \), \( b = -5 \)
So: \[ (x^2 - y^2)(-4) = 2xy(1 + 5) \Rightarrow -4(x^2 - y^2) = 12xy \Rightarrow x^2 + 3xy - y^2 = 0 \]