To find the ratio in which the line segment joining the points A(-2, 3, 6) and B(3, 4, -1) is divided by the plane 2x + 3y - z = 3, we can follow these steps:
### Step 1: Define the Points and the Plane
Let A(-2, 3, 6) and B(3, 4, -1) be the points. The equation of the plane is given as 2x + 3y - z = 3.
### Step 2: Assume the Ratio
Assume the line segment AB is divided by the point C in the ratio k:1. The coordinates of point C can be expressed using the section formula as follows:
- \( C_x = \frac{k \cdot x_2 + 1 \cdot x_1}{k + 1} \)
- \( C_y = \frac{k \cdot y_2 + 1 \cdot y_1}{k + 1} \)
- \( C_z = \frac{k \cdot z_2 + 1 \cdot z_1}{k + 1} \)
Where \( (x_1, y_1, z_1) = (-2, 3, 6) \) and \( (x_2, y_2, z_2) = (3, 4, -1) \).
### Step 3: Calculate the Coordinates of C
Substituting the coordinates of A and B into the formulas:
- \( C_x = \frac{k \cdot 3 + 1 \cdot (-2)}{k + 1} = \frac{3k - 2}{k + 1} \)
- \( C_y = \frac{k \cdot 4 + 1 \cdot 3}{k + 1} = \frac{4k + 3}{k + 1} \)
- \( C_z = \frac{k \cdot (-1) + 1 \cdot 6}{k + 1} = \frac{-k + 6}{k + 1} \)
### Step 4: Substitute C into the Plane Equation
Since point C lies on the plane, we substitute \( C_x, C_y, C_z \) into the plane equation:
\[ 2C_x + 3C_y - C_z = 3 \]
Substituting the values:
\[ 2\left(\frac{3k - 2}{k + 1}\right) + 3\left(\frac{4k + 3}{k + 1}\right) - \left(\frac{-k + 6}{k + 1}\right) = 3 \]
### Step 5: Simplify the Equation
Multiply through by \( k + 1 \) to eliminate the denominator:
\[ 2(3k - 2) + 3(4k + 3) + (k - 6) = 3(k + 1) \]
Expanding each term:
\[ 6k - 4 + 12k + 9 + k - 6 = 3k + 3 \]
Combine like terms:
\[ (6k + 12k + k) + (-4 + 9 - 6) = 3k + 3 \]
This simplifies to:
\[ 19k - 1 = 3k + 3 \]
### Step 6: Solve for k
Rearranging gives:
\[ 19k - 3k = 3 + 1 \]
\[ 16k = 4 \]
\[ k = \frac{1}{4} \]
### Step 7: Find the Ratio
The ratio in which the line segment is divided is \( k:1 \), which is:
\[ \frac{1}{4}:1 \]
This can be expressed as:
\[ 1:4 \]
### Final Answer
The ratio in which the line segment joining the points A and B is divided by the plane is **1:4**.
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