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 divided the line?

Options

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

Answer

To solve the problem, we need to find the ratio in which the plane \(2x + y - z = 3\) divides the line segment joining the points \(A(2, 1, 3)\) and \(B(4, -2, 5)\).

### Step-by-Step Solution:

1. **Find the Direction Ratios of the Line Segment:**
The direction ratios of the line segment joining points \(A(2, 1, 3)\) and \(B(4, -2, 5)\) can be calculated as follows:
\[ \text{Direction Ratios} = (4 - 2, -2 - 1, 5 - 3) = (2, -3, 2) \]

2. **Parametric Equations of the Line:**
Using the direction ratios, we can write the parametric equations of the line segment:
\[ x = 2 + 2t, \quad y = 1 - 3t, \quad z = 3 + 2t \]
where \(t\) varies from 0 to 1.

3. **Substituting into the Plane Equation:**
Substitute the parametric equations into the plane equation \(2x + y - z = 3\):
\[ 2(2 + 2t) + (1 - 3t) - (3 + 2t) = 3 \]
Simplifying this gives:
\[ 4 + 4t + 1 - 3t - 3 - 2t = 3 \]
\[ 4t - 2t - 3t + 4 + 1 - 3 = 3 \]
\[ -t + 2 = 3 \]
\[ -t = 1 \implies t = -1 \]

4. **Finding the Coordinates of the Intersection Point:**
Substitute \(t = -1\) back into the parametric equations to find the coordinates of the intersection point \(P\):
\[ x = 2 + 2(-1) = 0, \quad y = 1 - 3(-1) = 4, \quad z = 3 + 2(-1) = 1 \]
Thus, the coordinates of point \(P\) are \((0, 4, 1)\).

5. **Using the Section Formula to Find the Ratio:**
Let the ratio in which the plane divides the line segment \(AB\) be \(m:n\). According to the section formula:
\[ P\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}, \frac{mz_2 + nz_1}{m+n}\right) = (0, 4, 1) \]
Substituting the coordinates of \(A\) and \(B\):
- For \(x\):
\[ \frac{4m + 2n}{m+n} = 0 \implies 4m + 2n = 0 \implies 2n = -4m \implies \frac{m}{n} = -\frac{1}{2} \]
- For \(y\):
\[ \frac{-2m + 1n}{m+n} = 4 \implies -2m + n = 4(m+n) \implies -2m + n = 4m + 4n \implies -6m - 3n = 0 \]
- For \(z\):
\[ \frac{5m + 3n}{m+n} = 1 \implies 5m + 3n = m + n \implies 4m + 2n = 0 \]
This confirms our earlier result.

6. **Final Ratio:**
The ratio \(m:n\) is \(1:2\) (the negative sign indicates that the division is external).

### Conclusion:
The ratio in which the plane divides the line segment joining the points \(A(2, 1, 3)\) and \(B(4, -2, 5)\) is \(1:2\).