Distance between two non - intersecting planes `P_(1) and P_(2)` is 5 units, where `P_(1)` is `2x-3y+6z+26=0` and `P_(2)` is `4x+by+cz+d=0`. The point `A(-3, 0,-1)` lies between the planes `P_(1)` and `P_(2)`, then the value of `3b+4c-5d` is equal to

Options

A. 580
B. 120
C. `-18`
D. `-120`

Answer

To solve the problem step by step, we will follow the outlined approach:

### Step 1: Identify the planes
We have two planes:
- Plane \( P_1: 2x - 3y + 6z + 26 = 0 \)
- Plane \( P_2: 4x + by + cz + d = 0 \)

### Step 2: Determine if the planes are parallel
For the planes to be parallel, the coefficients of \( x \), \( y \), and \( z \) must be proportional.

The coefficients of \( P_1 \) are \( (2, -3, 6) \) and for \( P_2 \) they are \( (4, b, c) \).

Setting up the proportionality:
\[ \frac{2}{4} = \frac{-3}{b} = \frac{6}{c} \]

From \( \frac{2}{4} = \frac{1}{2} \):
1. \( \frac{-3}{b} = \frac{1}{2} \) gives \( b = -6 \).
2. \( \frac{6}{c} = \frac{1}{2} \) gives \( c = 12 \).

### Step 3: Calculate the distance between the planes
The distance \( d \) between two parallel planes \( Ax + By + Cz + D_1 = 0 \) and \( Ax + By + Cz + D_2 = 0 \) is given by:
\[ \text{Distance} = \frac{|D_2 - D_1|}{\sqrt{A^2 + B^2 + C^2}} \]

For our planes:
- \( D_1 = 26 \) (from \( P_1 \))
- \( D_2 = d \) (from \( P_2 \))
- The coefficients \( A = 2, B = -3, C = 6 \)

Calculating the denominator:
\[ \sqrt{2^2 + (-3)^2 + 6^2} = \sqrt{4 + 9 + 36} = \sqrt{49} = 7 \]

Setting the distance equal to 5 units:
\[ \frac{|d - 26|}{7} = 5 \]

### Step 4: Solve for \( d \)
Multiplying both sides by 7:
\[ |d - 26| = 35 \]

This gives two cases:
1. \( d - 26 = 35 \) → \( d = 61 \)
2. \( d - 26 = -35 \) → \( d = -9 \)

### Step 5: Calculate \( 3b + 4c - 5d \)
We have \( b = -6 \), \( c = 12 \), and two possible values for \( d \).

**Case 1: \( d = 61 \)**
\[ 3b + 4c - 5d = 3(-6) + 4(12) - 5(61) \]
Calculating:
\[ = -18 + 48 - 305 = -18 + 48 - 305 = -275 \]

**Case 2: \( d = -9 \)**
\[ 3b + 4c - 5d = 3(-6) + 4(12) - 5(-9) \]
Calculating:
\[ = -18 + 48 + 45 = -18 + 48 + 45 = 75 \]

### Conclusion
The two possible values of \( 3b + 4c - 5d \) are \( -275 \) and \( 75 \). However, the problem states that the point \( A(-3, 0, -1) \) lies between the planes, which implies we need to check which value of \( d \) is valid.