Find the ratio in which the join 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 solve the problem of finding the ratio in which the line segment joining points A (2, 1, 5) and B (3, 4, 3) is divided by the plane 2x + 2y - 2z = 1, we can follow these steps:

### Step 1: Define the Points
Let:
- Point A = (x1, y1, z1) = (2, 1, 5)
- Point B = (x2, y2, z2) = (3, 4, 3)

### Step 2: Assume the Ratio
Assume the point C divides the line segment AB in the ratio k:1. The coordinates of point C can be expressed using the section formula:
\[ C\left(\frac{kx2 + x1}{k + 1}, \frac{ky2 + y1}{k + 1}, \frac{kz2 + z1}{k + 1}\right) \]

### Step 3: Substitute the Coordinates
Substituting the coordinates of points A and B into the formula:
\[ C\left(\frac{k \cdot 3 + 2}{k + 1}, \frac{k \cdot 4 + 1}{k + 1}, \frac{k \cdot 3 + 5}{k + 1}\right) \]

### Step 4: Check Condition with the Plane Equation
Since point C lies on the plane described by the equation \(2x + 2y - 2z = 1\), we substitute the coordinates of C into the plane equation:
\[ 2\left(\frac{3k + 2}{k + 1}\right) + 2\left(\frac{4k + 1}{k + 1}\right) - 2\left(\frac{3k + 5}{k + 1}\right) = 1 \]

### Step 5: 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 6: Solve for k
Rearranging gives:
\[ 8k - k = 1 + 4 \implies 7k = 5 \implies k = \frac{5}{7} \]

### Step 7: Find the Ratio
Thus, the ratio in which the line segment AB is divided by the plane is:
\[ \text{Ratio} = k:1 = \frac{5}{7}:1 = 5:7 \]

### Step 8: Find the Coordinates of Point C
Now, substitute \(k = \frac{5}{7}\) back into the coordinates of C:
\[ C\left(\frac{3 \cdot \frac{5}{7} + 2}{\frac{5}{7} + 1}, \frac{4 \cdot \frac{5}{7} + 1}{\frac{5}{7} + 1}, \frac{3 \cdot \frac{5}{7} + 5}{\frac{5}{7} + 1}\right) \]

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

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

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

### Final Coordinates of Point C
Thus, the coordinates of point C are:
\[ C\left(\frac{29}{12}, \frac{27}{12}, \frac{50}{12}\right) \]

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