If 4x+ay=0, 5x+2y=0, bx+cy=0 and dx+4y=0 are the sides of a square find a, b, c and d.

In the question you have given the equations of the sides
as 4x+ay=0, 5x+2y=0, bx+cy=0 and dx+4y=0. I am going to take these as the equations of
adjoining sides.

Now the opposite sides of a square are
parallel to each other and the adjoining sides are perpendicular to each
other.

Let's convert all the lines to the slope and y-
intercept form:

4x+ay=0 => y= (-4 /a) x
....(1)

5x+2y=0 => y= (-5/2 ) x
....(2)

bx+cy=0 => y = (-b/c) x 
....(3)

and dx+4y=0 => y = (-d/4) x
....(4)

Now, slope of 1 = slope of
3,

=> -4/a =
-b/c

=> a= 4c/b

slope
of 2 = slope of 4

=> -5/2 =
-d/4

=> d = 5*4/2=
10

Also, 1 and 2 are perpendicular, so (-4 /a) = -1/ (-5/2
)

=> a/4 =
-5/2

=> a =-5*4/2 =
-10

And 3 and 4 are perpendicular, so –b/c= -1/
(-d/4)

=> c/b =
-d/4

=> d= -4*c/b

From
both a= 4c/b= -10 and d = -4 *c/b =10 we can only get that c = -10b/4 =
-5b/2

Therefore using the given equations we
can only determine d=10 and a=-10.

b and c can take any value
such that c= -5*b/2.