Key Concepts and Formulas

• Intercept Form of a Straight 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 $M$ of a line segment with endpoints $P(x_1, y_1)$ and $Q(x_2, y_2)$ is given by: $M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$

Step-by-Step Solution

Step 1: Define the line using the intercept form.

• We are given information about the x and y intercepts, so we will use the intercept form of a line.
• Let the equation of the straight line be $\frac{x}{a} + \frac{y}{b} = 1$.
• Here, $a$ is the x-intercept, so the line crosses the x-axis at the point $(a, 0)$.
• And $b$ is the y-intercept, so the line crosses the y-axis at the point $(0, b)$.

Step 2: Apply the given condition that A(3, 4) bisects the intercept.

• The problem states that the point $A(3, 4)$ bisects the line segment between the x and y intercepts. This means that $A(3, 4)$ is the midpoint of the line segment connecting the points $(a, 0)$ and $(0, b)$.
• We will use the midpoint formula to relate the coordinates of the intercepts to the coordinates of point $A$.

Step 3: Use the Midpoint Formula to set up equations for $a$ and $b$.

• Using the midpoint formula for the segment with endpoints $(a, 0)$ and $(0, b)$, the midpoint's coordinates are: $\left(\frac{a+0}{2}, \frac{0+b}{2}\right) = \left(\frac{a}{2}, \frac{b}{2}\right)$
• Since this midpoint is given as $A(3, 4)$, we can equate the corresponding coordinates: $\frac{a}{2} = 3 \quad \text{and} \quad \frac{b}{2} = 4$

Step 4: Solve for the x-intercept ($a$) and y-intercept ($b$).

• From the equation $\frac{a}{2} = 3$, we multiply both sides by 2 to solve for $a$: $a = 3 \times 2 \implies a = 6$
• From the equation $\frac{b}{2} = 4$, we multiply both sides by 2 to solve for $b$: $b = 4 \times 2 \implies b = 8$
• So, the x-intercept of the line is $6$, and the y-intercept is $8$.

Step 5: Substitute the values of $a$ and $b$ back into the intercept form of the line's equation.

• Now that we have $a=6$ and $b=8$, we can write the equation of the line as: $\frac{x}{6} + \frac{y}{8} = 1$

Step 6: Simplify the equation.

• To eliminate the denominators, we find the least common multiple (LCM) of $6$ and $8$. The LCM of $6$ and $8$ is $24$.
• Multiply the entire equation by $24$: $24 \left(\frac{x}{6}\right) + 24 \left(\frac{y}{8}\right) = 24 \times 1$ $4x + 3y = 24$

Step 7: Check if the point (3,4) satisfies the equation and compare with given options.

• Substitute x=3 and y=4 in the equation 4x+3y=24
• 4(3) + 3(4) = 12 + 12 = 24. Hence the point satisfies the equation.
• The equation matches option (C).

Common Mistakes & Tips

• Intercept Form: Recognizing when to use the intercept form is crucial. Look for keywords like "intercepts" or "cuts the axes."
• Midpoint Formula: Double-check that you're using the midpoint formula correctly and that you've assigned the coordinates correctly.
• Verification: Always verify your final equation by substituting the given point to ensure it lies on the line.

Summary

This problem demonstrates how to combine the intercept form of a line with the midpoint formula. By recognizing that the given point bisects the intercept between the axes, we can use the midpoint formula to determine the x and y intercepts. Substituting these values back into the intercept form gives us the equation of the line. The final equation is $4x + 3y = 24$, which corresponds to option (C).

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