# Find the equation of the line passes through midpoint A(2,4) and ( -1, -6) and parallel to the line 2y-4x - 8 = 0

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The line passes through midpoint AB :

Then let us find the point M such that M is midpoint AB:

Mx = ( MA+MB)/2 = ( 2+-1) / 2 = 1/2

My = ( yA+yB)/2 = ( -6+4)/2 = (-2/2) = -1

Then the point M(1/2, -1) passes through the line:

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

Given that the kine is parallel to 2y-4x -8 = 0

==> 2y = 4x + 8

==> y= 2x + 4

==> m = 2

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

==> y= 2x -1 -1

==>** y= 2x -2**

To find the line through the mid point of A(2,4) and B(-1,-6) and parallel to 2y-2x-4.

The mid point of AB = M(x,y) = {(Ax+Bx)/2 , (Ay+By)/2} = {(2-1)/2 . (4-6)/2} = (1/2 ,-1).

Any line through (x1,y1) = y-y1 = m(x-x1) where m is the slope of the line.

Since the line through (1/2, -1) is parallel to 2y-4x-8 = 0, has the slope of 2y-4x-8 = 0 which is m = -(coefficient of x)/(coefficient y) = -(-4)/2= 2.

Any line through the mid point of AB = (1/2 , -1) which is parallel to y has the slope = 2. So its equation should be:

y - 1/2 = 2{x- -(-1)} = 2x+2

y-1/2 = 2x+2. We multiply this equation by 2:

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

2y-2 = 4x+4.

2y-4x-2-4 = 0.

2y-4x -6 is parallel to 2y -4x-8 and passes through (1/2, -1) , the mid point of A and B.