A point P lies on the line whose end points are `A(1,2,3) and B(2,10,1)`. If z-co-ordinate of P is 7, then its other co-ordinate is ….......

Answer

To find the coordinates of point P that lies on the line segment AB with endpoints A(1, 2, 3) and B(2, 10, 1), given that the z-coordinate of P is 7, we can follow these steps:

### Step 1: Understand the coordinates of points A and B
The coordinates of point A are (1, 2, 3) and the coordinates of point B are (2, 10, 1).

### Step 2: Use the section formula
Point P divides the line segment AB in the ratio of λ:1. The coordinates of point P can be expressed using the section formula:
\[ P(x, y, z) = \left( \frac{m x_2 + n x_1}{m+n}, \frac{m y_2 + n y_1}{m+n}, \frac{m z_2 + n z_1}{m+n} \right) \]
where \( m = λ \) and \( n = 1 \).

### Step 3: Substitute the coordinates into the formula
Using the coordinates of A and B:
- \( x_1 = 1, y_1 = 2, z_1 = 3 \)
- \( x_2 = 2, y_2 = 10, z_2 = 1 \)

The coordinates of P become:
\[ P(x, y, z) = \left( \frac{λ \cdot 2 + 1 \cdot 1}{λ + 1}, \frac{λ \cdot 10 + 1 \cdot 2}{λ + 1}, \frac{λ \cdot 1 + 1 \cdot 3}{λ + 1} \right) \]

### Step 4: Set the z-coordinate equal to 7
Since we know the z-coordinate of P is 7, we set up the equation:
\[ \frac{λ \cdot 1 + 3}{λ + 1} = 7 \]

### Step 5: Solve for λ
Cross-multiplying gives:
\[ λ + 3 = 7(λ + 1) \]
Expanding this:
\[ λ + 3 = 7λ + 7 \]
Rearranging gives:
\[ 3 - 7 = 7λ - λ \]
\[ -4 = 6λ \]
Thus, we find:
\[ λ = -\frac{2}{3} \]

### Step 6: Substitute λ back into the coordinates of P
Now we substitute \( λ = -\frac{2}{3} \) back into the equations for the x and y coordinates of P:
1. For the x-coordinate:
\[ x = \frac{(-\frac{2}{3}) \cdot 2 + 1}{-\frac{2}{3} + 1} = \frac{-\frac{4}{3} + 1}{-\frac{2}{3} + 1} = \frac{-\frac{4}{3} + \frac{3}{3}}{-\frac{2}{3} + \frac{3}{3}} = \frac{-\frac{1}{3}}{\frac{1}{3}} = -1 \]

2. For the y-coordinate:
\[ y = \frac{(-\frac{2}{3}) \cdot 10 + 2}{-\frac{2}{3} + 1} = \frac{-\frac{20}{3} + 2}{-\frac{2}{3} + 1} = \frac{-\frac{20}{3} + \frac{6}{3}}{-\frac{2}{3} + \frac{3}{3}} = \frac{-\frac{14}{3}}{\frac{1}{3}} = -14 \]

### Step 7: Final coordinates of P
Thus, the coordinates of point P are:
\[ P(-1, -14, 7) \]