Let the line be:

(y-yA) = m(x-xA) where A(0,4) passes through the line:

==> (y-4) = m(x-0)

==> y= (m1)x + 4

Now we are givern that the line is perpendicular to the line 4x + 8y + 9 = 0

Then the product of their slopes = 1

Let us calculate the slope for the given line:

4x + 8y + 9 = 0

==> y= -(1/2)x - 9/8

==> the slope (m2) = -1/2

=> m1*m2 = -1

==> m1*-1/2 = -1

==> m1 = 2

==> y= m1x + 4 = 2x + 4

**==> y= 2x + 4**

To find the line perpendicular to 4x+8y+9 = 0 through the point (0,4).

Solution:

A line through (x1,y1) with a slope m is given by:

y = m(x-x1)+y1.

Any line through the point (0,4) with a slope m is given by:

y = m(x-0)+4 or

y = mx+4. We shall determine by the fact that this line is perpendicular to 4x+8y+ 9 = 0 or

8y = -4x-9 or

y = -(1/2)x+9/8, which is in slope intercept form.Therefore the slope (of 4x+8y+9) is -1/2.

The two lines y = mx+4 and y = -1/2x-9/8 are perpendicular only if the product of their slope is -1.

m*(1/2) = -1.

m = -2.

So the line y = mx+4 becomes y = 2x+4. Or

y-2x-4 is the line through (0,4) perpendicular to 4x+8y+9 = 0 is

y -2x-4 = 0 or 2x-y+4 = 0

Let's express 4x + 8y + 9 =0 in the form y=mx+b where m is the slope and b is the y-intercept.

4x + 8y + 9 =0

=> 8y = -4x -9

=> y = (-4/8)*x - (9/8)

Now if the slope of a line is m, and the slope of a perpendicular line is x m*x=-1 => x= -1/m.

Therefore the slope of the required line is -1/ (4/8) or (-8/4) = -2.

Let it be y= -2x + b

Now we know that this line passes through (0,4), therefore 4= -2*0+b or b= 4

The required line is y= -2x+4

**=> 2x+y-4=0**

The slope of the given line can be found out by converting the equation in the form y = mx + c. In this form of equation the slope of the line is m, and that of any line perpendicular to it is equal to -1/m.

Given equation of line:

4x + 8y + 9 = 0

==> 8y = - 4x - 9

==> y = - (4/8)x - 9/8

==> y = - x/2 - 9/8

Therefor:

Slope of given line = -1/2

And scope of line perpendicular to it:

= -1/(-1/2) = 2

Therefore equation of line perpendicular to given line will be:

y = 2x + c

To find the value of c, we substitute coordinated of the given point in above equation. Thus:

4 = 2*0 + c

==> 4 = c

Substituting this value of c in above equation of perpendicular we get:

y = 2x + 4

This equation can be simplified as:

2x - y + 4 = 0