Let `x/a + y/b + z/c = 1` be a plane, then the image of origin with respect to the plane is :

Options

A. `[(2)/(a ((1)/(a ^(2))+(1)/(b ^(2) ) + (1 )/(c ^(2)))), (2)/(b((1)/(a ^(2))+ (1)/(b ^(2))+ (1)/(c ^(2)))),(2)/(c ((1)/(a ^(2))+(1)/(b ^(2)) + (1)/(c ^(2))))]`
B. `((1)/((1)/(a ^(2)) + (1)/(b ^(2)) + (1)/(c ^(2))), (1)/((1)/(a ^(2))+ (1)/(b ^(2))+(1)/(c ^(2))),(1)/((1)/(a ^(2))+ (1)/(b^(2))+(1)/(c ^(2))))`
C. `(-2, -b, -2c)`
D. None of these

Explanation

To find the image of the origin with respect to the plane given by the equation \( \frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1 \), we can follow these steps:

### Step 1: Rewrite the Plane Equation
The given equation of the plane can be rewritten in the standard form:
\[ \frac{x}{a} + \frac{y}{b} + \frac{z}{c} - 1 = 0 \]
This can be expressed as:
\[ \frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1 \implies ax + by + cz = abc \]
where \( D = abc \).

### Step 2: Identify the Coefficients
From the plane equation \( ax + by + cz = abc \), we can identify the coefficients:
- \( A = a \)
- \( B = b \)
- \( C = c \)
- \( D = abc \)

### Step 3: Use the Image Formula
The formula for the image of a point \( (x_1, y_1, z_1) \) with respect to the plane \( Ax + By + Cz = D \) is given by:
\[ \left( x', y', z' \right) = \left( x_1 - \frac{2A(Ax_1 + By_1 + Cz_1 - D)}{A^2 + B^2 + C^2}, y_1 - \frac{2B(Ax_1 + By_1 + Cz_1 - D)}{A^2 + B^2 + C^2}, z_1 - \frac{2C(Ax_1 + By_1 + Cz_1 - D)}{A^2 + B^2 + C^2} \right) \]

### Step 4: Substitute the Origin Coordinates
For the origin, we have \( (x_1, y_1, z_1) = (0, 0, 0) \). Substituting these values into the formula:
\[ Ax_1 + By_1 + Cz_1 - D = 0 + 0 + 0 - abc = -abc \]
Thus, we can substitute into the image formula:
\[ x' = 0 - \frac{2a(-abc)}{a^2 + b^2 + c^2} = \frac{2abc}{a^2 + b^2 + c^2} \]
\[ y' = 0 - \frac{2b(-abc)}{a^2 + b^2 + c^2} = \frac{2abc}{a^2 + b^2 + c^2} \]
\[ z' = 0 - \frac{2c(-abc)}{a^2 + b^2 + c^2} = \frac{2abc}{a^2 + b^2 + c^2} \]

### Step 5: Final Result
Thus, the image of the origin with respect to the plane is:
\[ \left( \frac{2abc}{a^2 + b^2 + c^2}, \frac{2abc}{a^2 + b^2 + c^2}, \frac{2abc}{a^2 + b^2 + c^2} \right) \]