Find the ratio in which the line joint of A (2,1,5) and B(3,4,3) is divided by the plane `2x+ 2y - 2z = 1 ` . Also , find the coordinates of the point of division .

Answer

To find the ratio in which the line segment joining points A(2, 1, 5) and B(3, 4, 3) is divided by the plane given by the equation \(2x + 2y - 2z = 1\), we can follow these steps:

### Step 1: Determine the coordinates of point C
Let point C divide the line segment AB in the ratio \(k:1\). Using the section formula, the coordinates of point C can be expressed as:

\[ C\left(\frac{3k + 2}{k + 1}, \frac{4k + 1}{k + 1}, \frac{3k + 5}{k + 1}\right) \]

### Step 2: Substitute the coordinates of C into the plane equation
Since point C lies on the plane, we can substitute its coordinates into the plane equation \(2x + 2y - 2z = 1\):

\[ 2\left(\frac{3k + 2}{k + 1}\right) + 2\left(\frac{4k + 1}{k + 1}\right) - 2\left(\frac{3k + 5}{k + 1}\right) = 1 \]

### Step 3: Simplify the equation
Multiply through by \(k + 1\) to eliminate the denominator:

\[ 2(3k + 2) + 2(4k + 1) - 2(3k + 5) = k + 1 \]

Expanding this gives:

\[ 6k + 4 + 8k + 2 - 6k - 10 = k + 1 \]

Combining like terms results in:

\[ 8k - 4 = k + 1 \]

### Step 4: Solve for k
Rearranging the equation:

\[ 8k - k = 1 + 4 \]
\[ 7k = 5 \]
\[ k = \frac{5}{7} \]

### Step 5: Find the coordinates of point C
Now substitute \(k = \frac{5}{7}\) back into the coordinates of C:

\[ C\left(\frac{3(\frac{5}{7}) + 2}{\frac{5}{7} + 1}, \frac{4(\frac{5}{7}) + 1}{\frac{5}{7} + 1}, \frac{3(\frac{5}{7}) + 5}{\frac{5}{7} + 1}\right) \]

Calculating each coordinate:

1. For x-coordinate:
\[ x = \frac{\frac{15}{7} + 2}{\frac{5}{7} + 1} = \frac{\frac{15}{7} + \frac{14}{7}}{\frac{5}{7} + \frac{7}{7}} = \frac{\frac{29}{7}}{\frac{12}{7}} = \frac{29}{12} \]

2. For y-coordinate:
\[ y = \frac{\frac{20}{7} + 1}{\frac{5}{7} + 1} = \frac{\frac{20}{7} + \frac{7}{7}}{\frac{5}{7} + \frac{7}{7}} = \frac{\frac{27}{7}}{\frac{12}{7}} = \frac{27}{12} \]

3. For z-coordinate:
\[ z = \frac{\frac{15}{7} + 5}{\frac{5}{7} + 1} = \frac{\frac{15}{7} + \frac{35}{7}}{\frac{5}{7} + \frac{7}{7}} = \frac{\frac{50}{7}}{\frac{12}{7}} = \frac{50}{12} \]

### Final Coordinates of C
Thus, the coordinates of point C are:

\[ C\left(\frac{29}{12}, \frac{27}{12}, \frac{50}{12}\right) \]

### Conclusion
The ratio in which the line segment AB is divided by the plane is \(5:7\), and the coordinates of point C are \(\left(\frac{29}{12}, \frac{27}{12}, \frac{50}{12}\right)\).