Key Concepts and Formulas

• Intercept Form of a Line: A line with x-intercept $a$ and y-intercept $b$ has the equation $\frac{x}{a} + \frac{y}{b} = 1$.
• Midpoint Formula: The midpoint of a line segment with endpoints $(x_1, y_1)$ and $(x_2, y_2)$ is $\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$.

Step-by-Step Solution

1. Define the Intercepts and Points: Let the equation of the straight line in intercept form be $\frac{x}{a} + \frac{y}{b} = 1$ This line intersects the x-axis at point $A(a, 0)$ and the y-axis at point $B(0, b)$. The line segment $AB$ is the intercepted portion between the axes.

2. Apply the Midpoint Condition: The problem states that $P(-3, 4)$ bisects the line segment $AB$. This means $P$ is the midpoint of $AB$. We will use the midpoint formula to relate the coordinates of $A$, $B$, and $P$.

Applying the midpoint formula to $A(a, 0)$ and $B(0, b)$, the midpoint of $AB$ is: $\left( \frac{a+0}{2}, \frac{0+b}{2} \right) = \left( \frac{a}{2}, \frac{b}{2} \right)$ Since this midpoint is $P(-3, 4)$, we have: $\left( \frac{a}{2}, \frac{b}{2} \right) = (-3, 4)$

3. Solve for the Intercepts 'a' and 'b': Equating the coordinates, we get two equations: $\frac{a}{2} = -3 \quad \text{and} \quad \frac{b}{2} = 4$ Solving for $a$ and $b$: $a = 2 \times -3 = -6$ $b = 2 \times 4 = 8$ So, the x-intercept is $-6$ and the y-intercept is $8$.

4. Substitute Intercepts into the Equation of the Line: Substitute the values of $a = -6$ and $b = 8$ back into the intercept form of the line equation: $\frac{x}{-6} + \frac{y}{8} = 1$

5. Simplify the Equation to Standard Form: To eliminate the denominators, multiply the entire equation by the least common multiple (LCM) of $6$ and $8$, which is $24$: $24 \left( \frac{x}{-6} \right) + 24 \left( \frac{y}{8} \right) = 24 \times 1$ $-4x + 3y = 24$ Rearrange the terms to get the standard form $Ax + By + C = 0$: $4x - 3y + 24 = 0$

6. Verify the Solution with Options: Comparing our derived equation $4x - 3y + 24 = 0$ with the given options, we find that it matches option (B). Let's also confirm our answer with the given point. Substituting (-3,4) into the equation gives us: 4(-3) - 3(4) + 24 = -12 -12 + 24 = 0. This means that the point lies on the line.

Common Mistakes & Tips

• Choosing the Correct Form: Using the intercept form directly simplifies the problem. Starting with another form, such as point-slope, would require additional steps to relate the slope and point to the intercepts.
• Sign Errors: Pay close attention to signs when solving for the intercepts and substituting them back into the equation.
• LCM Multiplication: Make sure to multiply every term in the equation by the LCM when clearing denominators, including the constant term on the right side.

Summary

By recognizing that the given point is the midpoint of the line segment intercepted by the axes, and utilizing the intercept form of a line's equation, we efficiently solved for the intercepts and derived the equation of the line. The final answer is $4x - 3y + 24 = 0$, which corresponds to option (B).

The final answer is $\boxed{4x - 3y + 24 = 0}$, which corresponds to option (B).