Direction-ratios of line given by :
`(x -1)/(3) = (2y + 6)/(10) = (1 -z)/(-7) ` are ,

Options

A. `lt 3, 10, -7 gt`
B. `lt 3, -5, 7 gt`
C. `lt 3,5,7 gt`
D. `lt 3,5,-7 gt`,

Answer

To find the direction ratios of the line given by the equation

\[ \frac{x - 1}{3} = \frac{2y + 6}{10} = \frac{1 - z}{-7} \]

we will follow these steps:

### Step 1: Rewrite the equation
The given equation can be rewritten in a more standard form. We will express each part of the equation separately:

\[ \frac{x - 1}{3} = \frac{2y + 6}{10} = \frac{1 - z}{-7} \]

### Step 2: Simplify the second term
To simplify the second term, we can factor out the common factor from the numerator:

\[ \frac{2y + 6}{10} = \frac{2(y + 3)}{10} = \frac{y + 3}{5} \]

### Step 3: Rewrite the third term
For the third term, we can change the sign of the denominator:

\[ \frac{1 - z}{-7} = -\frac{1 - z}{7} = \frac{z - 1}{7} \]

### Step 4: Combine the terms
Now we can rewrite the entire equation as:

\[ \frac{x - 1}{3} = \frac{y + 3}{5} = \frac{z - 1}{7} \]

### Step 5: Identify the direction ratios
From the rewritten equation, we can identify the direction ratios directly from the denominators:

- The direction ratio corresponding to \(x\) is \(3\),
- The direction ratio corresponding to \(y\) is \(5\),
- The direction ratio corresponding to \(z\) is \(7\).

### Final Answer
Thus, the direction ratios of the line are:

\[ \text{Direction Ratios} = (3, 5, 7) \]

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