If a line has direction-cosines `lt (-9)/(11), (6)/(11), (-2)/(11)gt` , then what are its direction-ratios?

Answer

To find the direction ratios of a line given its direction cosines, we can follow these steps:

### Step 1: Understand the relationship between direction cosines and direction ratios.
Direction cosines (l, m, n) are the cosines of the angles that a line makes with the coordinate axes. Direction ratios (a, b, c) are proportional to the direction cosines. The relationship can be expressed as:
\[ l = \frac{a}{\sqrt{a^2 + b^2 + c^2}} \]
\[ m = \frac{b}{\sqrt{a^2 + b^2 + c^2}} \]
\[ n = \frac{c}{\sqrt{a^2 + b^2 + c^2}} \]

### Step 2: Given direction cosines.
The direction cosines provided are:
\[ l = -\frac{9}{11}, \quad m = \frac{6}{11}, \quad n = -\frac{2}{11} \]

### Step 3: Set up the equations for direction ratios.
Using the relationships from Step 1, we can express the direction ratios (a, b, c) in terms of a common factor k:
\[ l = \frac{a}{\sqrt{a^2 + b^2 + c^2}} \Rightarrow a = k \cdot l \]
\[ m = \frac{b}{\sqrt{a^2 + b^2 + c^2}} \Rightarrow b = k \cdot m \]
\[ n = \frac{c}{\sqrt{a^2 + b^2 + c^2}} \Rightarrow c = k \cdot n \]

Substituting the values of l, m, and n:
\[ a = k \cdot \left(-\frac{9}{11}\right) \]
\[ b = k \cdot \left(\frac{6}{11}\right) \]
\[ c = k \cdot \left(-\frac{2}{11}\right) \]

### Step 4: Find the common factor k.
To find k, we can use the property that the sum of the squares of the direction ratios should equal the square of the denominator of the direction cosines:
\[ \sqrt{a^2 + b^2 + c^2} = 11 \]
Thus:
\[ a^2 + b^2 + c^2 = 11^2 = 121 \]

Substituting the expressions for a, b, and c:
\[ \left(-\frac{9k}{11}\right)^2 + \left(\frac{6k}{11}\right)^2 + \left(-\frac{2k}{11}\right)^2 = 121 \]
\[ \frac{81k^2}{121} + \frac{36k^2}{121} + \frac{4k^2}{121} = 121 \]
\[ \frac{121k^2}{121} = 121 \]
\[ k^2 = 121 \Rightarrow k = 11 \quad (\text{taking the positive root}) \]

### Step 5: Calculate the direction ratios.
Now substituting k back into the equations for a, b, and c:
\[ a = 11 \cdot \left(-\frac{9}{11}\right) = -9 \]
\[ b = 11 \cdot \left(\frac{6}{11}\right) = 6 \]
\[ c = 11 \cdot \left(-\frac{2}{11}\right) = -2 \]

### Conclusion:
Thus, the direction ratios are:
\[ \text{Direction Ratios} = (-9, 6, -2) \]

---