The ratio in which the plane 3x+4y-5z =1 divided the join of (-2,4,-6) and (3,-5,6) is
(a)12:13 (b) 13:12 (c )13:14 (d)14:13

Answer

To solve the problem of finding the ratio in which the plane \(3x + 4y - 5z = 1\) divides the line segment joining the points \((-2, 4, -6)\) and \((3, -5, 6)\), we can follow these steps:

### Step 1: Identify the Points
Let the points be:
- \(A(-2, 4, -6)\)
- \(B(3, -5, 6)\)

### Step 2: Use Section Formula
Let \(P(x, y, z)\) be the point that divides the line segment \(AB\) in the ratio \(1:k\). According to the section formula, the coordinates of point \(P\) can be expressed as:
\[ P\left(\frac{-2k + 3}{k + 1}, \frac{4k - 5}{k + 1}, \frac{-6k + 6}{k + 1}\right) \]

### Step 3: Substitute into the Plane Equation
Since point \(P\) lies on the plane \(3x + 4y - 5z = 1\), we substitute the coordinates of \(P\) into the plane equation:
\[ 3\left(\frac{-2k + 3}{k + 1}\right) + 4\left(\frac{4k - 5}{k + 1}\right) - 5\left(\frac{-6k + 6}{k + 1}\right) = 1 \]

### Step 4: Simplify the Equation
Multiply through by \(k + 1\) to eliminate the denominator:
\[ 3(-2k + 3) + 4(4k - 5) - 5(-6k + 6) = k + 1 \]
Expanding this gives:
\[ -6k + 9 + 16k - 20 + 30k - 30 = k + 1 \]
Combining like terms:
\[ (40k - 41) = k + 1 \]

### Step 5: Solve for \(k\)
Rearranging the equation:
\[ 40k - k = 41 + 1 \]
\[ 39k = 42 \]
\[ k = \frac{42}{39} = \frac{14}{13} \]

### Step 6: Find the Ratio
Since the ratio is \(1:k\), we have:
\[ \text{Ratio} = 1 : \frac{14}{13} = 13 : 14 \]

### Final Answer
Thus, the ratio in which the plane divides the line segment is \(13:14\).

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