The equation of line equally inclined to coordinate axes and passing through (-3,2,-5) is

Options

A. `(x+3)/(1)=(y-2)/(1)=(z+5)/(1)`
B. `(x+3)/(-1)=(y-2)/(1)=(5+z)/(-1)`
C. `(x+3)/(-1)=(y-2)/(1)=(z+5)/(1)`
D. `(x+3)/(-1)=(2-y)/(1)=(z+5)/(-1)`

Explanation

To find the equation of the line that is equally inclined to the coordinate axes and passes through the point (-3, 2, -5), we can follow these steps:

### Step 1: Identify the point and direction cosines
We are given the point through which the line passes:
- \( P(-3, 2, -5) \)

Since the line is equally inclined to the coordinate axes, we can denote the direction cosines as:
- \( l = \pm 1 \)
- \( m = \pm 1 \)
- \( n = \pm 1 \)

For a line to be equally inclined to the axes, the direction ratios must be equal in magnitude. We can choose:
- \( l = -1 \) (for the x-axis)
- \( m = 1 \) (for the y-axis)
- \( n = -1 \) (for the z-axis)

### Step 2: Write the equation of the line
The general form of the equation of a line in three-dimensional space that passes through a point \( (x_1, y_1, z_1) \) with direction cosines \( (l, m, n) \) is given by:
\[ \frac{x - x_1}{l} = \frac{y - y_1}{m} = \frac{z - z_1}{n} \]

Substituting the values:
- \( x_1 = -3 \)
- \( y_1 = 2 \)
- \( z_1 = -5 \)
- \( l = -1 \)
- \( m = 1 \)
- \( n = -1 \)

We get:
\[ \frac{x + 3}{-1} = \frac{y - 2}{1} = \frac{z + 5}{-1} \]

### Step 3: Simplify the equation
This can be rewritten as:
\[ -(x + 3) = y - 2 = -(z + 5) \]

Thus, we can express it as:
1. \( -(x + 3) = y - 2 \)
2. \( -(x + 3) = -(z + 5) \)

From the first equation:
\[ y - 2 = -x - 3 \implies y = -x - 1 \]

From the second equation:
\[ -(x + 3) = -(z + 5) \implies x + 3 = z + 5 \implies z = x - 2 \]

### Final Equation
Therefore, the equations of the line can be expressed as:
\[ y = -x - 1 \quad \text{and} \quad z = x - 2 \]

### Summary
The equation of the line equally inclined to the coordinate axes and passing through the point (-3, 2, -5) is given by:
\[ \frac{x + 3}{-1} = \frac{y - 2}{1} = \frac{z + 5}{-1} \]