A ray of light comes light comes along the line L = 0 and strikes the plane mirror kept along the plane P = 0 at B. `A(2, 1, 6)` is a point on the line L = 0 whose image about P = 0 is `A'`. It is given that L = 0 is `(x-2)/(3)= (y-1)/(4)= (z-6)/(5) and P =0 ` is `x+y-2z=3`.
The coordinates of B are

Options

A. `(5, 10, 6)`
B. `(10, 15, 11)`
C. `(-10, -15, -14)`
D. none of these

Explanation

To find the coordinates of point B where the ray of light strikes the plane mirror, we will follow these steps:

### Step 1: Write the equations of the line and the plane
The line L is given by the parametric equations:
\[ \frac{x-2}{3} = \frac{y-1}{4} = \frac{z-6}{5} = \lambda \]
From this, we can express the coordinates of point B in terms of \(\lambda\):
- \(x = 3\lambda + 2\)
- \(y = 4\lambda + 1\)
- \(z = 5\lambda + 6\)

The equation of the plane P is given by:
\[ x + y - 2z = 3 \]

### Step 2: Substitute the coordinates of B into the plane equation
Substituting the expressions for \(x\), \(y\), and \(z\) into the plane equation:
\[ (3\lambda + 2) + (4\lambda + 1) - 2(5\lambda + 6) = 3 \]

### Step 3: Simplify the equation
Now, simplify the equation:
\[ 3\lambda + 2 + 4\lambda + 1 - 10\lambda - 12 = 3 \]
Combine like terms:
\[ (3\lambda + 4\lambda - 10\lambda) + (2 + 1 - 12) = 3 \]
This simplifies to:
\[ -3\lambda - 9 = 3 \]

### Step 4: Solve for \(\lambda\)
Now, isolate \(\lambda\):
\[ -3\lambda = 3 + 9 \]
\[ -3\lambda = 12 \]
\[ \lambda = -4 \]

### Step 5: Find the coordinates of point B
Now substitute \(\lambda = -4\) back into the equations for \(x\), \(y\), and \(z\):
- \(x = 3(-4) + 2 = -12 + 2 = -10\)
- \(y = 4(-4) + 1 = -16 + 1 = -15\)
- \(z = 5(-4) + 6 = -20 + 6 = -14\)

Thus, the coordinates of point B are:
\[ B(-10, -15, -14) \]

### Final Answer
The coordinates of point B are \((-10, -15, -14)\).

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