Question 430682
y = 8x-8 is in slope-intercept form, y=mx+b.
slope = 8
y-intercept is where x=0, which is -8.
.
A line perpendicular to a given line has a slope that is the inverse reciprocal.
Given the slope of the first line = 8, then a perpendicular line will have slope = -1/8.
,
We want this line to pass through (2,4). Its equation is:
y = -1/8x + b
.
Hmmm.  We don't know b.  (We can guess it would not be -8.)
.
But we can plug in the x & y we know (2,4) to determine what b has to be.
4 = -1/8(2) + b
4 = -1/4 + b
4 = -.25 + b
4.25 = b
So the equation of the perpendicular line is:
y = -1/8x + 4.25
.
I'm not sure what your last question about the parallel line references.  Two parallel lines that go through the same point will lie on top of one another.  Since a line has no third dimension, you cannot tell them apart.
.
But we can find a line parallel to the original line that goes through (2,4).
y = 8x -8 is the original equation.  But we don't care about the -8, because the intercept will shift to somewhere else for the parallel line.
y = 8x +b
4 = 8(2) +b
4 = 16 +b
-12 = b
So the equation of the parallel line is:
y = 8x -12
.
We can check our answers with a graph.
.
{{{graph(500,500,-10,10,-10,10,8*x-8, (-1/8)*x+4.25,8*x-12)}}}
.
So the answers are:
A line perpendicular to the first line that passes through (2,4) is defined by the equation:
y = -1/8x + 4.25  (This is the green line.)
A line parallel to the first line that passes through (2,4) is defined by the equation:
y = 8x -12 (This is the blue line.)
.
Done.