To find the ratio in which point [tex]\( P \)[/tex] partitions the directed line segment from [tex]\( M \)[/tex] to [tex]\( N \)[/tex], given that distance from [tex]\( M \)[/tex] to [tex]\( P \)[/tex] is [tex]\( \frac{9}{11} \)[/tex] of the total distance [tex]\( M \)[/tex] to [tex]\( N \)[/tex], follow these steps:
1. Interpret the Given Fraction:
Point [tex]\( P \)[/tex] divides the segment [tex]\( MN \)[/tex] such that [tex]\( MP \)[/tex] (distance from [tex]\( M \)[/tex] to [tex]\( P \)[/tex]) is [tex]\( \frac{9}{11} \)[/tex] of the total distance [tex]\( MN \)[/tex].
2. Calculate Remaining Distance:
To find the distance [tex]\( PN \)[/tex] (the distance from [tex]\( P \)[/tex] to [tex]\( N \)[/tex]), note that the entire distance is represented as 1 (the whole segment).
Hence, [tex]\( PN = 1 - MP = 1 - \frac{9}{11} = \frac{2}{11} \)[/tex].
3. Form the Ratio:
The ratio is derived by comparing [tex]\( MP \)[/tex] with [tex]\( PN \)[/tex].
So, we take [tex]\(\frac{MP}{PN}\)[/tex]:
[tex]\[
\frac{MP}{PN} = \frac{\frac{9}{11}}{\frac{2}{11}} = \frac{9}{2}
\][/tex]
4. Express the Ratio in the Simplest Form:
The ratio [tex]\( \frac{9}{2} \)[/tex] is equivalent to [tex]\( 9:2 \)[/tex].
Given these steps, the ratio in which point [tex]\( P \)[/tex] partitions the line segment [tex]\( MN \)[/tex] is [tex]\( 9:2 \)[/tex].
Thus, the correct answer is:
[tex]\[ 9:2 \][/tex]
