Key Concepts and Formulas

• Homogeneous Equation of a Pair of Straight Lines: $ax^2 + 2hxy + by^2 = 0$ represents a pair of straight lines passing through the origin.
• Equation of Angle Bisectors: The combined equation of the angle bisectors of the lines represented by $ax^2 + 2hxy + by^2 = 0$ is given by $\frac{x^2 - y^2}{a - b} = \frac{xy}{h}$, provided $a \neq b$ and $h \neq 0$. An alternative form that handles cases when $a=b$ or $h=0$ is $h(x^2 - y^2) = (a - b)xy$.

Step-by-Step Solution

1. Identify Coefficients

We are given the equation $x^2 - 4xy - 5y^2 = 0$. We need to identify the coefficients $a$, $h$, and $b$ by comparing it with the general form $ax^2 + 2hxy + by^2 = 0$.

By comparing the coefficients:

• Coefficient of $x^2$: $a = 1$
• Coefficient of $xy$: $2h = -4$, which means $h = -2$
• Coefficient of $y^2$: $b = -5$

Explanation: The coefficients are extracted directly from the given equation by comparing it to the standard form. Special attention is paid to the $xy$ term to correctly determine the value of $h$.

2. Apply the Angle Bisector Formula

We will use the formula for the combined equation of angle bisectors: $\frac{x^2 - y^2}{a - b} = \frac{xy}{h}$ Substitute the values of $a$, $h$, and $b$ into the formula: $\frac{x^2 - y^2}{1 - (-5)} = \frac{xy}{-2}$

Explanation: This step involves direct substitution of the values determined in the previous step into the angle bisector formula.

3. Simplify the Equation

Simplify the denominator: $\frac{x^2 - y^2}{6} = \frac{xy}{-2}$ Cross-multiply to eliminate the fractions: $-2(x^2 - y^2) = 6xy$ Distribute the $-2$: $-2x^2 + 2y^2 = 6xy$

Explanation: Basic algebraic simplification is performed to remove the fractions and prepare the equation for rearrangement into standard form.

4. Rearrange into Standard Form

Move all terms to one side to set the equation to zero: $0 = 2x^2 + 6xy - 2y^2$ Rewrite the equation: $2x^2 + 6xy - 2y^2 = 0$ Divide the entire equation by 2 to simplify: $x^2 + 3xy - y^2 = 0$

Explanation: The equation is rearranged into the standard homogeneous quadratic form. Dividing by the common factor of 2 simplifies the equation while preserving the represented pair of lines.

Common Pitfalls & Tips:

• Correctly identify h: Remember that the coefficient of the $xy$ term is $2h$, not $h$.
• Sign Errors: Pay close attention to negative signs during substitution and simplification.
• Simplification: Always simplify the equation by dividing out common factors.

Summary

By identifying the coefficients from the given equation and applying the formula for the equation of angle bisectors, we simplified and rearranged the resulting equation to obtain the equation of the pair of angle bisectors as $x^2 + 3xy - y^2 = 0$. This equation corresponds to option (C).

Final Answer

The final answer is \boxed{x^2 + 3xy - y^2 = 0}, which corresponds to option (C).