To determine the ratio in which point [tex]\( P \)[/tex] partitions the directed line segment from [tex]\( M \)[/tex] to [tex]\( N \)[/tex], given that point [tex]\( P \)[/tex] is [tex]\(\frac{9}{11}\)[/tex] of the distance from [tex]\( M \)[/tex] to [tex]\( N \)[/tex], follow these steps:
1. Understand the positioning of [tex]\( P \)[/tex]:
- The distance from [tex]\( M \)[/tex] to [tex]\( N \)[/tex] can be considered as a whole unit, which is 1.
- Point [tex]\( P \)[/tex] is located at [tex]\(\frac{9}{11}\)[/tex] of this distance from [tex]\( M \)[/tex] to [tex]\( N \)[/tex].
2. Calculate the remaining distance:
- If point [tex]\( P \)[/tex] is [tex]\(\frac{9}{11}\)[/tex] of the way from [tex]\( M \)[/tex] to [tex]\( N \)[/tex], the remaining distance from [tex]\( P \)[/tex] to [tex]\( N \)[/tex] is the fraction of the unit distance that's left, which can be calculated as:
[tex]\[
1 - \frac{9}{11} = \frac{11}{11} - \frac{9}{11} = \frac{2}{11}.
\][/tex]
3. Establish the ratio:
- The ratio in which [tex]\( P \)[/tex] partitions the segment can now be expressed as the ratio of the distance from [tex]\( M \)[/tex] to [tex]\( P \)[/tex] to the distance from [tex]\( P \)[/tex] to [tex]\( N \)[/tex].
- This can be written as:
[tex]\[
\frac{\frac{9}{11}}{\frac{2}{11}}.
\][/tex]
4. Simplify the ratio:
- The common denominator (11) can be cancelled out in the fraction, simplifying the ratio:
[tex]\[
\frac{9}{2}.
\][/tex]
- Therefore, the ratio can be expressed as [tex]\( 9:2 \)[/tex].
So, the point [tex]\( P \)[/tex] divides the line segment [tex]\( MN \)[/tex] into the ratio [tex]\( 9:2 \)[/tex].
Hence, the correct answer is:
[tex]\[
\boxed{9:2}
\][/tex]
