A(-2,2) and C(4,-1)are opposite vertices of a parallelogram ABCD whose sides are parallel to the lines x=0 and 3y=x, a)Find the coordinates of B...

Given the points A(-2, 2) and c(4,-1) are opposite
vertices's of a parallelogram.

Then the equations of the
lines for the parallelogram can be obtained .

A(-2,2) is on
both lines, so it satisfies both equations.

==> y-2
= m (x+2)

Now the slope for one side is the same slope for
the line x= 0

==> Then, one side of the
parallelogram is parallel to the y-axis.

==> Then,
the equation of the line is

 x =
-2.............(1)

The other side of the parallelogram is
parallel to 3y=x ==> Then the slope is
(1/3)

==> y-2 =
(1/3)(x+2)

==> y= (1/3)x + 2/3 +
2

==> y= (1/3)x +8/3
............(2)

Then, we have the equation of the lines of
the sides of the parallelogram that intersects at the point
(-2,2)

Now we will find the other
sides.

==> We will use the point
B(4,-1)

==> The first equation of the line that is
parallel to the y-axis.

==> Then the equation is x =
4 ..........(3)

==> The other side has the following
equation.

==> y +1 = (1/3)
(x-4)

==> y= (1/3)x - 4/3
-1

==> y= (1/3)x -7/3
..............(4)

Now the remaining points are the
intersection points between the lines (1) with (4) and the lines (2) with
(3).

Intersection point between equation (1) and
(4):

==> x= -2  and y= (1/3)x -
7/3

==> y= (-2/3 - 7/3 = -9/3 =
-3

==> Then, one of the points is ( -2,
-3)

Intersection between lines (2) and
(3).

==> x= 4  and y= (1/3)x +8/3  ==> y= 4/3
+ 8/3 = 12/3 = 4

Then, the point is
(4,4)

==> Then, the points B and D are
(4,4) and (-2,-3).