The line joining the points (2, 1, 3) and (4, -2, 5) cuts the plane `2x+y-z=3`.
What is the ratio in which the plane divideds the line ?

Options

A. `1: 1`
B. `2 :3`
C. `3:4`
D. None of these

Explanation

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

### Step 1: Find the direction ratios of the line segment AB
The direction ratios of the line segment joining points \(A(2, 1, 3)\) and \(B(4, -2, 5)\) can be calculated as follows:

\[ \text{Direction ratios} = (x_2 - x_1, y_2 - y_1, z_2 - z_1) = (4 - 2, -2 - 1, 5 - 3) = (2, -3, 2) \]

### Step 2: Parametric equations of the line segment
Using the direction ratios, we can write the parametric equations of the line segment as:

\[ x = 2 + 2t, \quad y = 1 - 3t, \quad z = 3 + 2t \]

where \(t\) varies from \(0\) to \(1\).

### Step 3: Substitute the parametric equations into the plane equation
We substitute the parametric equations into the plane equation \(2x + y - z = 3\):

\[ 2(2 + 2t) + (1 - 3t) - (3 + 2t) = 3 \]

Expanding this:

\[ 4 + 4t + 1 - 3t - 3 - 2t = 3 \]

Combining like terms:

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

This simplifies to:

\[ -1 = 3 \]

### Step 4: Solve for \(t\)
Rearranging gives:

\[ 4t - 3t - 2t + 2 = 3 \]

This simplifies to:

\[ -t + 2 = 3 \implies -t = 1 \implies t = -1 \]

### Step 5: Find the coordinates of the intersection point
Substituting \(t = -1\) back into the parametric equations:

\[ x = 2 + 2(-1) = 0, \quad y = 1 - 3(-1) = 4, \quad z = 3 + 2(-1) = 1 \]

Thus, the intersection point is \(P(0, 4, 1)\).

### Step 6: Use the section formula to find the ratio
Let the ratio in which the plane divides the line segment be \(k:1\). According to the section formula:

\[ P = \left( \frac{k \cdot x_2 + 1 \cdot x_1}{k + 1}, \frac{k \cdot y_2 + 1 \cdot y_1}{k + 1}, \frac{k \cdot z_2 + 1 \cdot z_1}{k + 1} \right) \]

Substituting the coordinates of points \(A\) and \(B\):

\[ (0, 4, 1) = \left( \frac{4k + 2}{k + 1}, \frac{-2k + 1}{k + 1}, \frac{5k + 3}{k + 1} \right) \]

### Step 7: Set up equations for each coordinate
From the x-coordinate:

\[ 0 = \frac{4k + 2}{k + 1} \implies 4k + 2 = 0 \implies k = -\frac{1}{2} \]

From the y-coordinate:

\[ 4 = \frac{-2k + 1}{k + 1} \implies 4(k + 1) = -2k + 1 \implies 4k + 4 = -2k + 1 \implies 6k = -3 \implies k = -\frac{1}{2} \]

From the z-coordinate:

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

### Conclusion: The ratio in which the plane divides the line
Thus, the ratio in which the plane divides the line segment \(AB\) is \(1:2\) (since \(k = -\frac{1}{2}\) indicates an external division).