If a straight line passing through the point P(-3,4) is such that its intercepted portion between the coordinate axes is bisected a P, then its equation is

Options

A. x-y+7=0
B. 3x-4y+25=0
C. 4x+3y=0
D. 4x-3y+24=0

Answer

To find the equation of the straight line that passes through the point P(-3, 4) and has its intercepted portion between the coordinate axes bisected at point P, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Intercepted Portion**:
Let the x-intercept of the line be A(a, 0) and the y-intercept be B(0, b). The line will intercept the x-axis at point A and the y-axis at point B.

2. **Using the Midpoint Formula**:
Since point P(-3, 4) bisects the segment AB, we can use the midpoint formula. The midpoint M of a segment with endpoints A(x1, y1) and B(x2, y2) is given by:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Here, we have:
\[ M = P(-3, 4) = \left( \frac{a + 0}{2}, \frac{0 + b}{2} \right) \]

3. **Setting Up the Equations**:
From the midpoint formula, we can equate the coordinates:
\[ -3 = \frac{a}{2} \quad \text{and} \quad 4 = \frac{b}{2} \]

4. **Solving for a and b**:
From the first equation:
\[ a = -6 \]
From the second equation:
\[ b = 8 \]

5. **Writing the Equation of the Line**:
The equation of a line in intercept form is given by:
\[ \frac{x}{a} + \frac{y}{b} = 1 \]
Substituting the values of a and b:
\[ \frac{x}{-6} + \frac{y}{8} = 1 \]

6. **Clearing the Denominators**:
To eliminate the fractions, we can multiply through by the least common multiple (LCM) of the denominators, which is 24:
\[ 24 \left( \frac{x}{-6} + \frac{y}{8} \right) = 24 \]
This simplifies to:
\[ -4x + 3y = 24 \]

7. **Rearranging the Equation**:
Rearranging gives us:
\[ 4x - 3y + 24 = 0 \]

### Final Answer:
The equation of the line is:
\[ 4x - 3y + 24 = 0 \]