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

Options

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

Explanation

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

### Step 1: Understand the Intercept Form
The equation of a line in intercept form is given by:
\[ \frac{x}{a} + \frac{y}{b} = 1 \]
where \( a \) is the x-intercept and \( b \) is the y-intercept.

### Step 2: Set Up the Problem
Since the point \( P(-3, 4) \) bisects the intercepted portion between the axes, we can denote the x-intercept as \( A(a, 0) \) and the y-intercept as \( B(0, b) \). The midpoint \( P \) of segment \( AB \) can be calculated as:
\[ P = \left( \frac{a + 0}{2}, \frac{0 + b}{2} \right) = \left( \frac{a}{2}, \frac{b}{2} \right) \]
Given that \( P = (-3, 4) \), we can equate:
\[ \frac{a}{2} = -3 \quad \text{and} \quad \frac{b}{2} = 4 \]

### Step 3: Solve for \( a \) and \( b \)
From the equations above, we can solve for \( a \) and \( b \):
1. From \( \frac{a}{2} = -3 \):
\[ a = -6 \]
2. From \( \frac{b}{2} = 4 \):
\[ b = 8 \]

### Step 4: Substitute \( a \) and \( b \) into the Intercept Form
Now we substitute \( a \) and \( b \) back into the intercept form of the line:
\[ \frac{x}{-6} + \frac{y}{8} = 1 \]

### Step 5: Clear the Denominators
To eliminate the fractions, we can multiply the entire equation by \( -24 \) (the least common multiple of the denominators):
\[ -24 \left( \frac{x}{-6} \right) + -24 \left( \frac{y}{8} \right) = -24 \]
This simplifies to:
\[ 4x - 3y = -24 \]

### Step 6: Rearranging the Equation
Rearranging gives us:
\[ 4x - 3y + 24 = 0 \]

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