To determine the ratio in which point [tex]\( P \)[/tex] divides the directed line segment from [tex]\( M \)[/tex] to [tex]\( N \)[/tex], given that [tex]\( P \)[/tex] is [tex]\(\frac{9}{11}\)[/tex] of the distance from [tex]\( M \)[/tex] to [tex]\( N \)[/tex], let's take the following steps:
1. Understanding the Given Information:
- Point [tex]\( P \)[/tex] is [tex]\(\frac{9}{11}\)[/tex] of the distance from [tex]\( M \)[/tex] to [tex]\( N \)[/tex].
- This means point [tex]\( P \)[/tex] is located at a position such that the length from [tex]\( M \)[/tex] to [tex]\( P \)[/tex] is [tex]\(\frac{9}{11}\)[/tex] of the total distance from [tex]\( M \)[/tex] to [tex]\( N \)[/tex].
2. Calculate the Remaining Segment:
- 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:
[tex]\[
1 - \frac{9}{11} = \frac{11}{11} - \frac{9}{11} = \frac{2}{11}
\][/tex]
3. Express the Distances as Ratios:
- The segment from [tex]\( M \)[/tex] to [tex]\( P \)[/tex] is [tex]\(\frac{9}{11}\)[/tex] of the total segment length.
- The segment from [tex]\( P \)[/tex] to [tex]\( N \)[/tex] is [tex]\(\frac{2}{11}\)[/tex] of the total segment length.
4. Convert to a Ratio:
- Since point [tex]\( P \)[/tex] divides [tex]\( M \)[/tex] to [tex]\( N \)[/tex] into two segments, one from [tex]\( M \)[/tex] to [tex]\( P \)[/tex] and the other from [tex]\( P \)[/tex] to [tex]\( N \)[/tex], we can express this as a ratio of the lengths of these segments, which are [tex]\( 9 \)[/tex] parts to [tex]\( 2 \)[/tex] parts.
- Therefore, the ratio is [tex]\( 9:2 \)[/tex].
So, the 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]
