The ratio in which the plane `vecr.(veci-2 vecj+3veck)=17` divides the line joining the points `-2veci+4vecj+7veckandvec3i-5vecj+8veck` is

Options

A. `1:5`
B. `1:10`
C. `3:5`
D. `3:10`

Explanation

To solve the problem, we need to find the ratio in which the plane \(\vec{r} \cdot (\vec{i} - 2\vec{j} + 3\vec{k}) = 17\) divides the line segment joining the points \(-2\vec{i} + 4\vec{j} + 7\vec{k}\) and \(3\vec{i} - 5\vec{j} + 8\vec{k}\).

### Step 1: Identify the Points
Let point A be \(-2\vec{i} + 4\vec{j} + 7\vec{k}\) and point B be \(3\vec{i} - 5\vec{j} + 8\vec{k}\).

### Step 2: Use the Section Formula
We denote the point C, which divides the line segment AB in the ratio \( \lambda : 1 \). According to the section formula, the coordinates of point C can be expressed as:
\[ C = \left( \frac{3\lambda - 2}{\lambda + 1}, \frac{-5\lambda + 4}{\lambda + 1}, \frac{8\lambda + 7}{\lambda + 1} \right) \]

### Step 3: Substitute C into the Plane Equation
The coordinates of point C must satisfy the plane equation:
\[ \vec{r} \cdot (\vec{i} - 2\vec{j} + 3\vec{k}) = 17 \]
This translates to:
\[ x - 2y + 3z = 17 \]
Substituting the coordinates of C into this equation:
\[ \frac{3\lambda - 2}{\lambda + 1} - 2\left(\frac{-5\lambda + 4}{\lambda + 1}\right) + 3\left(\frac{8\lambda + 7}{\lambda + 1}\right) = 17 \]

### Step 4: Simplify the Equation
Multiply through by \(\lambda + 1\) to eliminate the denominator:
\[ 3\lambda - 2 + 10\lambda - 8 + 24\lambda + 21 = 17(\lambda + 1) \]
Combine like terms:
\[ (3\lambda + 10\lambda + 24\lambda) - 2 - 8 + 21 = 17\lambda + 17 \]
This simplifies to:
\[ 37\lambda + 11 = 17\lambda + 17 \]

### Step 5: Solve for \(\lambda\)
Rearranging gives:
\[ 37\lambda - 17\lambda = 17 - 11 \]
\[ 20\lambda = 6 \implies \lambda = \frac{6}{20} = \frac{3}{10} \]

### Step 6: Find the Ratio
The ratio in which the plane divides the line segment AB is:
\[ \lambda : 1 = \frac{3}{10} : 1 = 3 : 10 \]

Thus, the final answer is that the plane divides the line segment in the ratio \(3 : 10\).

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