What point on the y-axis is 12sqrt13/13 units from the line defined by equation 3x -2y + 6 = 0?Answer: (0,-3) (0,9)

Given the equation of the
line:

3x - 2y + 6 = 0

We need
to find the point ( x,y) such that the distance between the line and the point is
12sqrt13

We will use thedistance between a point and a
line.

D = l ax + by + c l / sqrt(a^2 +
b^2)

==> 12sqrt13/ 13 = l 3x -2y + 6 l /
sqrt(9+4)

==> 12sqrt13 / 13 = l 3x-2y + 6 l /
sqrt13

Multiply and divide the right side by
sqrt13.

==> 12sqrt13 = l 3x-2y + 6l sqrt13/
13

==> l 3x -2y + 6 l =
12

==> But the point is on the y-axis, then x=
0

==> l -2y + 6 l =
12

Then, there are two
cases:

1) ==> -2y + 6 = 12 ==> -2y = 6
==> y= -3

2) ==> -(-2y+6) = 12 ==> 2y
= 18==> y= 9

Then the points are ( 0,
-3) and (0, 9) .