If the distance between A(2,-4) and B( 4,b ) is 12 units. Find b.

Given the point A(2,-4) and B(4, b) such that the distance
between them is 12 units.

We will use the distance between
two points formula to find the value of b.

==> We
know that:

D = sqrt( x1-x2)^2 +
(y1-y2)^2

==> sqrt( 4-2)^2 + (b+4)^2 =
12

==> sqrt( 4 + b^2 + 8b + 16) =
12

==> sqrt(b^2 + 8b + 20) =
12

Now we will square both
sides.

==> b^2 + 8b + 20 =
144

==> b^2 + 8b + 20 - 144 =
0

==> b^2 + 8b -124 =
0

Now we will find the
roots.

==> b1= ( -8 + sqrt( 64+496)
/2

            = ( -8 + 4sqrt35) /2 = (
-4+2sqrt35

==> b2=
-4-2sqrt35

Then there are two possible values for b such
that the distance between A and B is 12
.

==> b = { -4+2sqrt35  ,  -4-2sqrt35
}