To solve the problem of determining how the point [tex]\( P \)[/tex], which is [tex]\(\frac{9}{11}\)[/tex] of the distance from [tex]\( M \)[/tex] to [tex]\( N \)[/tex], partitions the directed line segment from [tex]\( M \)[/tex] to [tex]\( N \)[/tex] into a ratio, we can look at it step by step.
1. Determine the Fraction of Distance Covered:
- Point [tex]\( P \)[/tex] is given to be [tex]\(\frac{9}{11}\)[/tex] of the distance from [tex]\( M \)[/tex] to [tex]\( N \)[/tex]. This means that [tex]\( P \)[/tex] has covered [tex]\(\frac{9}{11}\)[/tex] of the total distance from [tex]\( M \)[/tex] to [tex]\( N \)[/tex].
2. Determine the Remaining Distance:
- Since the total distance from [tex]\( M \)[/tex] to [tex]\( N \)[/tex] is represented by a whole (1), the remaining distance from [tex]\( P \)[/tex] to [tex]\( N \)[/tex] would be:
[tex]\[
1 - \frac{9}{11}
\][/tex]
- Subtracting the covered distance:
[tex]\[
1 - \frac{9}{11} = \frac{11}{11} - \frac{9}{11} = \frac{2}{11}
\][/tex]
- Thus, the remaining distance from [tex]\( P \)[/tex] to [tex]\( N \)[/tex] is [tex]\(\frac{2}{11}\)[/tex] of the total distance.
3. Convert to a Ratio:
- We need to express the distance from [tex]\( M \)[/tex] to [tex]\( P \)[/tex] and the distance from [tex]\( P \)[/tex] to [tex]\( N \)[/tex] as a ratio.
- The distance from [tex]\( M \)[/tex] to [tex]\( P \)[/tex] is [tex]\(\frac{9}{11}\)[/tex].
- The distance from [tex]\( P \)[/tex] to [tex]\( N \)[/tex] is [tex]\(\frac{2}{11}\)[/tex].
4. Simplify the Ratio:
- Let's use the fraction values to form the ratio:
[tex]\[
\frac{M \text{ to } P}{P \text{ to } N} = \frac{\frac{9}{11}}{\frac{2}{11}}
\][/tex]
- Since the denominators are the same, they cancel out, leaving:
[tex]\[
\frac{9}{2}
\][/tex]
- Thus, the ratio is [tex]\( 9:2 \)[/tex].
5. Final Conclusion:
- Therefore, point [tex]\( P \)[/tex] partitions the directed line segment from [tex]\( M \)[/tex] to [tex]\( N \)[/tex] into the ratio [tex]\( 9:2 \)[/tex].
The correct answer is:
[tex]\[ \boxed{9:2} \][/tex]
