A line passing through the origin perpendicularly cuts the line 3x−2y=6 at point M. Find the coordinates of point M.

1. Find the slope of the given line:
The equation of the given line is:

3x−2y=6⟹2y=3x−6⟹y=
2
3
​
x−3
The slope of this line (m
1
​
) is
2
3
​
.

Find the equation of the perpendicular line passing through the origin (0,0):
The slope of a line perpendicular to it (m
2
​
) satisfies m
1
​
×m
2
​
=−1:

m
2
​
=−
m
1
​

1
​
=−
3
2
​

Since it passes through the origin, its equation is:

y=−
3
2
​
x⟹2x+3y=0
Find the intersection point M by solving the two equations simultaneously:
From the second equation, y=−
3
2
​
x. Substitute this into the first equation (3x−2y=6):

3x−2(−
3
2
​
x)=6
3x+
3
4
​
x=6
3
13
​
x=6⟹x=
13
18
​

Now, find the corresponding value of y:

y=−
3
2
​
(
13
18
​
)=−
13
12
​

Thus, the coordinates of point M are (
13
18
​
,−
13
12
​
).