Find the ratio in which the first point divides the join of other two: (0,-1,-7), (2,1,-9) and (6,5,- 13).

Answer

To find the ratio in which the first point \( A(0, -1, -7) \) divides the line segment joining the points \( B(2, 1, -9) \) and \( C(6, 5, -13) \), we will use the section formula.

### Step 1: Identify the points
Let:
- \( A = (0, -1, -7) \)
- \( B = (2, 1, -9) \)
- \( C = (6, 5, -13) \)

### Step 2: Use the section formula
The section formula states that if a point \( A(x_1, y_1, z_1) \) divides the line segment joining points \( B(x_2, y_2, z_2) \) and \( C(x_3, y_3, z_3) \) in the ratio \( m:n \), then:

\[ x_1 = \frac{mx_3 + nx_2}{m+n}, \quad y_1 = \frac{my_3 + ny_2}{m+n}, \quad z_1 = \frac{mz_3 + nz_2}{m+n} \]

### Step 3: Set up equations based on coordinates
We need to find \( m:n \) such that:

1. For the x-coordinates:
\[ 0 = \frac{m \cdot 6 + n \cdot 2}{m+n} \]

2. For the y-coordinates:
\[ -1 = \frac{m \cdot 5 + n \cdot 1}{m+n} \]

3. For the z-coordinates:
\[ -7 = \frac{m \cdot (-13) + n \cdot (-9)}{m+n} \]

### Step 4: Solve the equations

#### Solving the x-coordinate equation:
From the x-coordinate equation:
\[ 0 = m \cdot 6 + n \cdot 2 \implies 6m + 2n = 0 \implies 3m + n = 0 \implies n = -3m \]

#### Solving the y-coordinate equation:
Substituting \( n = -3m \) into the y-coordinate equation:
\[ -1 = \frac{m \cdot 5 + (-3m) \cdot 1}{m + (-3m)} = \frac{5m - 3m}{m - 3m} = \frac{2m}{-2m} = -1 \]
This equation holds true for any \( m \neq 0 \).

#### Solving the z-coordinate equation:
Substituting \( n = -3m \) into the z-coordinate equation:
\[ -7 = \frac{m \cdot (-13) + (-3m) \cdot (-9)}{m + (-3m)} = \frac{-13m + 27m}{m - 3m} = \frac{14m}{-2m} = -7 \]
This equation also holds true for any \( m \neq 0 \).

### Step 5: Determine the ratio
Since \( n = -3m \), we can express the ratio \( m:n \) as:
\[ m : n = m : (-3m) = 1 : -3 \]
However, since we are interested in the positive ratio, we take the absolute value:
\[ m : n = 1 : 3 \]

### Final Answer
The ratio in which point \( A(0, -1, -7) \) divides the line segment joining points \( B(2, 1, -9) \) and \( C(6, 5, -13) \) is \( 1:3 \).

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