The equation of a line passing through the point (-3 ,2 ,-4) and equally inclined to the axis are

Options

A. x-3=y+2=z-4
B. x+3=y-2=z+4
C. `(x+3)/(2)=(y-2)/(2)=(z-4)/(3)`
D. None of these

Answer

To find the equation of a line passing through the point (-3, 2, -4) and equally inclined to the axes, we can follow these steps:

### Step 1: Understand the concept of direction cosines
The direction cosines of a line are the cosines of the angles that the line makes with the coordinate axes. If a line is equally inclined to all three axes, then the direction cosines will be equal.

### Step 2: Set up the equation for direction cosines
Let the direction cosines be denoted as \( l \), \( m \), and \( n \). Since the line is equally inclined to the axes, we have:
\[ l = m = n \]

### Step 3: Use the property of direction cosines
From the property of direction cosines, we know:
\[ l^2 + m^2 + n^2 = 1 \]
Substituting \( l = m = n \) into this equation gives:
\[ 3l^2 = 1 \]
Thus,
\[ l^2 = \frac{1}{3} \]
Taking the square root, we find:
\[ l = m = n = \frac{1}{\sqrt{3}} \]

### Step 4: Write the equation of the line
The general form of the equation of a line in space can be expressed as:
\[ \frac{x - x_1}{l} = \frac{y - y_1}{m} = \frac{z - z_1}{n} \]
Substituting the point (-3, 2, -4) for \( (x_1, y_1, z_1) \) and \( l = m = n = \frac{1}{\sqrt{3}} \):
\[ \frac{x + 3}{\frac{1}{\sqrt{3}}} = \frac{y - 2}{\frac{1}{\sqrt{3}}} = \frac{z + 4}{\frac{1}{\sqrt{3}}} \]

### Step 5: Simplify the equation
Multiplying through by \( \sqrt{3} \) to eliminate the fractions gives:
\[ \sqrt{3}(x + 3) = \sqrt{3}(y - 2) = \sqrt{3}(z + 4) \]
This can be rewritten as:
\[ x + 3 = y - 2 = z + 4 \]

### Final Answer
The equation of the line is:
\[ x + 3 = y - 2 = z + 4 \]