A straight line through the point P(3,4) is such that its intercept between the axes is bisected at P. its equation is :

Options

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

Answer

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

### Step 1: Understand the intercepts
Let the x-intercept be \( H \) and the y-intercept be \( K \). The coordinates of the x-intercept are \( (H, 0) \) and the coordinates of the y-intercept are \( (0, K) \).

### Step 2: Find the midpoint of the intercepts
The midpoint of the intercepts can be calculated using the midpoint formula:
\[ \text{Midpoint} = \left( \frac{H + 0}{2}, \frac{0 + K}{2} \right) = \left( \frac{H}{2}, \frac{K}{2} \right) \]
Since this midpoint is given to be the point P(3, 4), we can set up the equations:
\[ \frac{H}{2} = 3 \quad \text{and} \quad \frac{K}{2} = 4 \]

### Step 3: Solve for H and K
From the equations above, we can solve for \( H \) and \( K \):
\[ H = 2 \times 3 = 6 \quad \text{and} \quad K = 2 \times 4 = 8 \]

### Step 4: Write the equation of the line
The equation of a line in intercept form is given by:
\[ \frac{x}{H} + \frac{y}{K} = 1 \]
Substituting the values of \( H \) and \( K \):
\[ \frac{x}{6} + \frac{y}{8} = 1 \]

### Step 5: Clear the denominators
To eliminate the fractions, we can multiply through by the least common multiple of the denominators (which is 24):
\[ 24 \left( \frac{x}{6} + \frac{y}{8} \right) = 24 \cdot 1 \]
This simplifies to:
\[ 4x + 3y = 24 \]

### Step 6: Rearranging the equation
We can rearrange this equation to standard form:
\[ 4x + 3y - 24 = 0 \]

Thus, the equation of the straight line is:
\[ 4x + 3y = 24 \]