To solve the problem, let's analyze the data given:
1. Understanding the distance partition:
- Point [tex]\( P \)[/tex] is [tex]\(\frac{9}{11}\)[/tex] of the distance from [tex]\( M \)[/tex] to [tex]\( N \)[/tex].
2. Visual Representation:
- Suppose the total distance from [tex]\( M \)[/tex] to [tex]\( N \)[/tex] is [tex]\( d \)[/tex].
- Distance from [tex]\( M \)[/tex] to [tex]\( P \)[/tex] is [tex]\(\frac{9}{11}d\)[/tex].
- Distance from [tex]\( P \)[/tex] to [tex]\( N \)[/tex] is [tex]\( d - \frac{9}{11}d \)[/tex].
3. Simplify the remaining distance:
- Distance from [tex]\( P \)[/tex] to [tex]\( N \)[/tex] is [tex]\( d - \frac{9}{11}d \)[/tex].
- [tex]\( P \)[/tex] to [tex]\( N \)[/tex] = [tex]\( d \left(1 - \frac{9}{11}\right) \)[/tex].
- Simplify the expression: [tex]\[ d \left(1 - \frac{9}{11}\right) = d \left(\frac{11}{11} - \frac{9}{11}\right) = d \left(\frac{2}{11}\right) = \frac{2d}{11}. \][/tex]
4. Establish the ratio [tex]\( M \)[/tex] to [tex]\( P \)[/tex] and [tex]\( P \)[/tex] to [tex]\( N \)[/tex]:
- Distance [tex]\( M \)[/tex] to [tex]\( P \)[/tex] = [tex]\(\frac{9d}{11}\)[/tex].
- Distance [tex]\( P \)[/tex] to [tex]\( N \)[/tex] = [tex]\(\frac{2d}{11}\)[/tex].
5. Forming the ratio:
- The ratio of the segments [tex]\( M \)[/tex] to [tex]\( P \)[/tex] and [tex]\( P \)[/tex] to [tex]\( N \)[/tex] is:
[tex]\[ \frac{\frac{9d}{11}}{\frac{2d}{11}} = \frac{9}{2}. \][/tex]
6. Conclusion:
- The ratio that [tex]\( P \)[/tex] partitions the line segment [tex]\( M \)[/tex] to [tex]\( N \)[/tex] into is [tex]\( 9:2 \)[/tex].
Hence, the correct answer is [tex]\(\boxed{9:2}\)[/tex].
