Key Concepts and Formulas

• Equation of Angle Bisectors: The equations of the angle bisectors between two lines $A_1x + B_1y + C_1 = 0$ and $A_2x + B_2y + C_2 = 0$ are given by: $\frac{A_1x + B_1y + C_1}{\sqrt{A_1^2 + B_1^2}} = \pm \frac{A_2x + B_2y + C_2}{\sqrt{A_2^2 + B_2^2}}$
• General Equation of a Pair of Straight Lines: The general equation of a pair of straight lines is given by: $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$
• Distance Formula: The distance of a point $(x, y)$ from a line $Ax + By + C = 0$ is given by: $d = \frac{|Ax + By + C|}{\sqrt{A^2 + B^2}}$

Step-by-Step Solution

Step 1: Find the equations of the angle bisectors.

We are given the two lines: $L_1: x + 2y + 7 = 0$ $L_2: 2x - y + 8 = 0$

Using the formula for the angle bisectors, we have: $\frac{x + 2y + 7}{\sqrt{1^2 + 2^2}} = \pm \frac{2x - y + 8}{\sqrt{2^2 + (-1)^2}}$ $\frac{x + 2y + 7}{\sqrt{5}} = \pm \frac{2x - y + 8}{\sqrt{5}}$ $x + 2y + 7 = \pm (2x - y + 8)$

Step 2: Find the two angle bisector equations.

Case 1: $x + 2y + 7 = 2x - y + 8$ $x - 3y + 1 = 0$

Case 2: $x + 2y + 7 = -(2x - y + 8)$ $x + 2y + 7 = -2x + y - 8$ $3x + y + 15 = 0$

Step 3: Combine the angle bisector equations to form a single equation.

The combined equation of the pair of straight lines is: $(x - 3y + 1)(3x + y + 15) = 0$ $3x^2 + xy + 15x - 9xy - 3y^2 - 45y + 3x + y + 15 = 0$ $3x^2 - 3y^2 - 8xy + 18x - 44y + 15 = 0$

Step 4: Compare the derived equation with the given equation.

We are given the equation: $x^2 - y^2 + 2hxy + 2gx + 2fy + c = 0$ Multiply this equation by 3 to match the coefficient of $x^2$ in our derived equation: $3x^2 - 3y^2 + 6hxy + 6gx + 6fy + 3c = 0$

Now, compare the coefficients: $6h = -8 \implies h = -\frac{4}{3}$ $6g = 18 \implies g = 3$ $6f = -44 \implies f = -\frac{22}{3}$ $3c = 15 \implies c = 5$

Step 5: Calculate the value of g + c + h - f.

$g + c + h - f = 3 + 5 - \frac{4}{3} - \left(-\frac{22}{3}\right)$ $= 8 - \frac{4}{3} + \frac{22}{3}$ $= 8 + \frac{18}{3}$ $= 8 + 6 = 14$

Step 6: Note the error in the original solution.

The original solution seems to have an error. The correct value for $g + c + h - f$ is 14, but it incorrectly jumps to 29.

Common Mistakes & Tips

• Sign Errors: Be very careful with signs when expanding and comparing coefficients. A small sign error can lead to a completely wrong answer.
• Normalization: Make sure to normalize the equations by dividing or multiplying by a constant to match the coefficients before comparing. This avoids errors due to scaling.
• Check the Arithmetic: Double-check all arithmetic calculations, especially when dealing with fractions.

Summary

We found the equations of the angle bisectors of the given lines, combined them into a single equation representing the pair of straight lines, and then compared the coefficients with the given general equation to find the values of $g$, $c$, $h$, and $f$. Finally, we calculated $g + c + h - f$ to be 14.

Final Answer

The final answer is \boxed{14}, which corresponds to option (B).