If `4x+4y-cz =0` is the equation of the plane passing through the origin that contains the line `(x+5)/(2)=(y)/(3)=(z-7)/(4)` then find the value of c

Answer

To find the value of \( c \) in the equation of the plane \( 4x + 4y - cz = 0 \) that passes through the origin and contains the line given by the equation \( \frac{x+5}{2} = \frac{y}{3} = \frac{z-7}{4} \), we can follow these steps:

### Step 1: Identify the Direction Ratios of the Line
The line can be expressed in parametric form. From the equation \( \frac{x+5}{2} = \frac{y}{3} = \frac{z-7}{4} \), we can set a parameter \( t \):
- \( x + 5 = 2t \) → \( x = 2t - 5 \)
- \( y = 3t \)
- \( z - 7 = 4t \) → \( z = 4t + 7 \)

From this, we can extract the direction ratios of the line:
- Direction ratios: \( (2, 3, 4) \)

### Step 2: Identify the Normal Vector of the Plane
The equation of the plane is given by \( 4x + 4y - cz = 0 \). The normal vector \( \mathbf{n} \) of this plane can be expressed as:
- Normal vector: \( (4, 4, -c) \)

### Step 3: Use the Perpendicularity Condition
Since the line lies in the plane, the direction ratios of the line must be perpendicular to the normal vector of the plane. This means their dot product must equal zero:
\[ (4, 4, -c) \cdot (2, 3, 4) = 0 \]

### Step 4: Calculate the Dot Product
Calculating the dot product:
\[ 4 \cdot 2 + 4 \cdot 3 + (-c) \cdot 4 = 0 \]
\[ 8 + 12 - 4c = 0 \]

### Step 5: Solve for \( c \)
Now, we can simplify the equation:
\[ 20 - 4c = 0 \]
\[ 4c = 20 \]
\[ c = \frac{20}{4} = 5 \]

### Final Answer
Thus, the value of \( c \) is \( 5 \).

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