The ratio in which the plane `2x-1=0` divides the line joining `(-2,4,7)` and `(3,-5,8)` is

Options

A. `2:3`
B. `4:5`
C. `7:8`
D. `1:1`

Explanation

To find the ratio in which the plane \(2x - 1 = 0\) divides the line joining the points \((-2, 4, 7)\) and \((3, -5, 8)\), we can follow these steps:

### Step 1: Identify the points
Let point A be \((-2, 4, 7)\) and point B be \((3, -5, 8)\).

### Step 2: Use the section formula
Assume the ratio in which the plane divides the line segment AB is \(k:1\). According to the section formula, the coordinates of the point that divides the line segment joining points A and B in the ratio \(k:1\) are given by:

\[ \left( \frac{3k - 2}{k + 1}, \frac{-5k + 4}{k + 1}, \frac{8k + 7}{k + 1} \right) \]

### Step 3: Set up the equation of the plane
The equation of the plane is given as \(2x - 1 = 0\). This can be rearranged to find \(x\):

\[ x = \frac{1}{2} \]

### Step 4: Substitute the x-coordinate into the plane equation
Substituting the x-coordinate from the section formula into the plane equation:

\[ \frac{3k - 2}{k + 1} = \frac{1}{2} \]

### Step 5: Cross-multiply to solve for k
Cross-multiplying gives:

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

Expanding both sides:

\[ 6k - 4 = k + 1 \]

### Step 6: Rearranging the equation
Rearranging the equation to isolate \(k\):

\[ 6k - k = 1 + 4 \]

\[ 5k = 5 \]

### Step 7: Solve for k
Dividing both sides by 5:

\[ k = 1 \]

### Step 8: Determine the ratio
The ratio in which the plane divides the line segment is \(k:1\), which is \(1:1\).

### Final Answer
The ratio in which the plane \(2x - 1 = 0\) divides the line joining the points \((-2, 4, 7)\) and \((3, -5, 8)\) is \(1:1\).

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