The ratio in which the join of the points `A(2,1,5)` and `B(3,4,3)` is divided by the plane `2x+2y-2z=1`, is

Options

A. `7:5`
B. `5:7`
C. `5:3`
D. `3:5`

Explanation

To find the ratio in which the line segment joining the points \( A(2, 1, 5) \) and \( B(3, 4, 3) \) is divided by the plane \( 2x + 2y - 2z = 1 \), we can use the section formula and the equation of the plane.

### Step 1: Write the coordinates of the point dividing the segment in the ratio \( \lambda : 1 \)

Let the point \( C \) divide the segment \( AB \) in the ratio \( \lambda : 1 \). The coordinates of point \( C \) can be expressed using the section formula:

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

### Step 2: Substitute the coordinates of point \( C \) into the plane equation

The coordinates of point \( C \) must satisfy the equation of the plane \( 2x + 2y - 2z = 1 \). Substituting the coordinates of \( C \):

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

### Step 3: Simplify the equation

Multiply through by \( \lambda + 1 \) to eliminate the denominator:

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

Expanding this gives:

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

Combine like terms:

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

### Step 4: Solve for \( \lambda \)

Rearranging the equation:

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

This simplifies to:

\[ 7\lambda = 5 \implies \lambda = \frac{5}{7} \]

### Step 5: Find the ratio

The ratio in which the line segment \( AB \) is divided by the plane is \( \lambda : 1 \), which is:

\[ \frac{5}{7} : 1 \]

Thus, the final ratio is:

\[ \frac{5}{7} : 1 \quad \text{or} \quad 5 : 7 \]