Find the co-ordinates of the point at a distance of `sqrt(14)` from the mid-point of AB on the line joining the point `A(1,2,3)` and `B(3,6,9)`.

Answer

To find the coordinates of the point at a distance of \( \sqrt{14} \) from the midpoint of points A and B, we can follow these steps:

### Step 1: Find the Midpoint of AB
The coordinates of points A and B are given as:
- \( A(1, 2, 3) \)
- \( B(3, 6, 9) \)

The formula for the midpoint \( M \) of a line segment joining two points \( A(x_1, y_1, z_1) \) and \( B(x_2, y_2, z_2) \) is:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \right) \]

Substituting the coordinates of A and B:
\[ M = \left( \frac{1 + 3}{2}, \frac{2 + 6}{2}, \frac{3 + 9}{2} \right) = \left( \frac{4}{2}, \frac{8}{2}, \frac{12}{2} \right) = (2, 4, 6) \]

### Step 2: Find the Direction Ratios of Line AB
The direction ratios of the line joining points A and B can be calculated as follows:
\[ \text{Direction Ratios} = (x_2 - x_1, y_2 - y_1, z_2 - z_1) = (3 - 1, 6 - 2, 9 - 3) = (2, 4, 6) \]

### Step 3: Parametric Equations of Line AB
Using the direction ratios, we can write the parametric equations of the line:
\[ x = 1 + 2\lambda, \quad y = 2 + 4\lambda, \quad z = 3 + 6\lambda \]

### Step 4: Find the Distance from Midpoint M
We need to find the points on the line that are at a distance of \( \sqrt{14} \) from the midpoint \( M(2, 4, 6) \). The distance \( d \) between two points \( P(x, y, z) \) and \( M(2, 4, 6) \) is given by:
\[ d = \sqrt{(x - 2)^2 + (y - 4)^2 + (z - 6)^2} \]
Setting \( d = \sqrt{14} \):
\[ \sqrt{(x - 2)^2 + (y - 4)^2 + (z - 6)^2} = \sqrt{14} \]
Squaring both sides:
\[ (x - 2)^2 + (y - 4)^2 + (z - 6)^2 = 14 \]

### Step 5: Substitute Parametric Equations into Distance Formula
Substituting the parametric equations into the distance formula:
\[ (1 + 2\lambda - 2)^2 + (2 + 4\lambda - 4)^2 + (3 + 6\lambda - 6)^2 = 14 \]
This simplifies to:
\[ (2\lambda - 1)^2 + (4\lambda - 2)^2 + (6\lambda - 3)^2 = 14 \]

### Step 6: Expand and Simplify
Expanding each term:
\[ (2\lambda - 1)^2 = 4\lambda^2 - 4\lambda + 1 \]
\[ (4\lambda - 2)^2 = 16\lambda^2 - 16\lambda + 4 \]
\[ (6\lambda - 3)^2 = 36\lambda^2 - 36\lambda + 9 \]
Combining these:
\[ 4\lambda^2 - 4\lambda + 1 + 16\lambda^2 - 16\lambda + 4 + 36\lambda^2 - 36\lambda + 9 = 14 \]
This simplifies to:
\[ 56\lambda^2 - 56\lambda + 14 = 14 \]
Subtracting 14 from both sides:
\[ 56\lambda^2 - 56\lambda = 0 \]
Factoring out \( 56\lambda \):
\[ 56\lambda(\lambda - 1) = 0 \]

### Step 7: Solve for \( \lambda \)
This gives us two solutions:
\[ \lambda = 0 \quad \text{or} \quad \lambda = 1 \]

### Step 8: Find Coordinates for Each \( \lambda \)
1. For \( \lambda = 0 \):
\[ x = 1 + 2(0) = 1, \quad y = 2 + 4(0) = 2, \quad z = 3 + 6(0) = 3 \quad \Rightarrow (1, 2, 3) \]

2. For \( \lambda = 1 \):
\[ x = 1 + 2(1) = 3, \quad y = 2 + 4(1) = 6, \quad z = 3 + 6(1) = 9 \quad \Rightarrow (3, 6, 9) \]

### Conclusion
The coordinates of the points at a distance of \( \sqrt{14} \) from the midpoint \( M(2, 4, 6) \) on the line joining points A and B are:
- \( (1, 2, 3) \)
- \( (3, 6, 9) \)