The coordinates of the point on the directed line segment from (2, -6) to (6, 2) that partitions the segment into a ratio of 3 to 5 are (-5, -6).
To find this point, we can use the following formula:
P
=
5
3
+
5
A
+
3
3
+
5
B
P=
3+5
5
A+
3+5
3
B
where
A
=
(
2
,
−
6
)
A=(2,−6) and
B
=
(
6
,
2
)
B=(6,2) are the endpoints of the line segment, and
P
P is the point we are looking for.
Plugging in the values, we get:
P
=
5
8
(
2
,
−
6
)
+
3
8
(
6
,
2
)
=
(
−
5
,
−
6
)
P=
8
5
(2,−6)+
8
3
(6,2)=(−5,−6)
Therefore, the coordinates of the point on the directed line segment from (2, -6) to (6, 2) that partitions the segment into a ratio of 3 to 5 are (-5, -6).