The lines `(x-x_1)/(a) = (y-y_1)/(b) = (z-z_1)/( c ) and (x-x_1)/(a') = (y-y_1)/(b') = (z-z_1)/( c')` are (i) parallel (ii) intersecting (iii) skew (iv) coincident

Options

A. parallel
B. intersecting
C. skew
D. coincident

Answer

To determine the relationship between the two lines given by the equations:

1. \(\frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c}\) (Line 1)
2. \(\frac{x - x_1}{a'} = \frac{y - y_1}{b'} = \frac{z - z_1}{c'}\) (Line 2)

we will analyze their direction ratios and points of intersection.

### Step 1: Identify the direction ratios and points of the lines

For Line 1:
- Direction ratios: \((a, b, c)\)
- Point on the line: \((x_1, y_1, z_1)\)

For Line 2:
- Direction ratios: \((a', b', c')\)
- Point on the line: \((x_1, y_1, z_1)\)

### Step 2: Check the direction ratios

The direction ratios of the two lines are different if:
\[ \frac{a}{a'} \neq \frac{b}{b'} \neq \frac{c}{c'} \]
If the ratios are equal, the lines are either coincident or parallel.

### Step 3: Check if the lines pass through the same point

Both lines pass through the point \((x_1, y_1, z_1)\). Since they share a common point, we need to analyze their direction ratios further.

### Step 4: Determine the relationship between the lines

Since the lines share the same point but have different direction ratios, they cannot be parallel or coincident. Therefore, they must be intersecting.

### Conclusion

The two lines are intersecting at the point \((x_1, y_1, z_1)\).

Thus, the correct option is:
**(ii) intersecting.**

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