Let's work on understanding the given situation step-by-step.
1. Understanding the Distance Ratio:
Point [tex]\( P \)[/tex] is located [tex]\(\frac{9}{11}\)[/tex] of the way from point [tex]\( M \)[/tex] to point [tex]\( N \)[/tex]. This means that [tex]\( P \)[/tex] divides the segment [tex]\( MN \)[/tex] into two parts.
2. Calculating the Remaining Distance:
Since the total distance from [tex]\( M \)[/tex] to [tex]\( N \)[/tex] is 1 (when considering the whole segment), the remaining part of the distance is:
[tex]\[
1 - \frac{9}{11}
\][/tex]
This simplifies to:
[tex]\[
1 - \frac{9}{11} = \frac{11}{11} - \frac{9}{11} = \frac{2}{11}
\][/tex]
3. Establishing the Ratio:
The point [tex]\( P \)[/tex] divides the segment into two parts:
- From [tex]\( M \)[/tex] to [tex]\( P \)[/tex] is [tex]\(\frac{9}{11}\)[/tex] of the distance.
- From [tex]\( P \)[/tex] to [tex]\( N \)[/tex] is [tex]\(\frac{2}{11}\)[/tex] of the distance.
4. Forming the Partition Ratio:
To find the partition ratio, compare the distances:
[tex]\[
\frac{9}{11} \text{ (from M to P)} \text{ and } \frac{2}{11} \text{ (from P to N)}
\][/tex]
The ratio of [tex]\( \frac{9}{11} \)[/tex] to [tex]\( \frac{2}{11} \)[/tex] is simply:
[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]. Thus, the correct answer is:
[tex]\[ 9:2 \][/tex]
