If the line joining A(1,3,4) and B is divided by the point (-2,3,5) in the ratio 1:3 then the coordinates of B is

Options

A. ` ( -11,3,8) `
B. ` ( -11,8,3)`
C. ` ( 11,-8,3)`
D. ` (-11,-3,8)`

Answer

To find the coordinates of point B, we will use the section formula. The section formula states that if a point C divides the line segment joining points A(x1, y1, z1) and B(x2, y2, z2) in the ratio m:n, then the coordinates of point C can be calculated as follows:

\[ C\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}, \frac{mz_2 + nz_1}{m+n}\right) \]

Given:
- Point A = (1, 3, 4)
- Point C = (-2, 3, 5)
- The ratio m:n = 1:3

Let the coordinates of point B be (x, y, z).

### Step 1: Set up the equations using the section formula
Using the section formula, we can express the coordinates of point C as:

\[ C\left(\frac{1 \cdot x + 3 \cdot 1}{1 + 3}, \frac{1 \cdot y + 3 \cdot 3}{1 + 3}, \frac{1 \cdot z + 3 \cdot 4}{1 + 3}\right) \]

This simplifies to:

\[ C\left(\frac{x + 3}{4}, \frac{y + 9}{4}, \frac{z + 12}{4}\right) \]

### Step 2: Equate the coordinates of point C to the given point (-2, 3, 5)
Now we can set up the equations by equating the coordinates:

1. \(\frac{x + 3}{4} = -2\)
2. \(\frac{y + 9}{4} = 3\)
3. \(\frac{z + 12}{4} = 5\)

### Step 3: Solve for x
From the first equation:

\[ \frac{x + 3}{4} = -2 \]

Multiply both sides by 4:

\[ x + 3 = -8 \]

Subtract 3 from both sides:

\[ x = -8 - 3 = -11 \]

### Step 4: Solve for y
From the second equation:

\[ \frac{y + 9}{4} = 3 \]

Multiply both sides by 4:

\[ y + 9 = 12 \]

Subtract 9 from both sides:

\[ y = 12 - 9 = 3 \]

### Step 5: Solve for z
From the third equation:

\[ \frac{z + 12}{4} = 5 \]

Multiply both sides by 4:

\[ z + 12 = 20 \]

Subtract 12 from both sides:

\[ z = 20 - 12 = 8 \]

### Final Answer
Thus, the coordinates of point B are:

\[ B(-11, 3, 8) \]