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).
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
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
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