Answer: Choice C) x-y-8 = 0
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How to get this answer:
First find the equation of the line through the origin (0,0) and (4,-4)
slope:
m = (y2-y1)/(x2-x1)
m = (-4-0)/(4-0)
m = -4/4
m = -1
Slope intercept form:
y = mx+b
y = -1*x+b
-4 = -1*4+b
-4 = -4+b
-4+4 = -4+b+4
0 = b
b = 0
The equation of the line through (0,0) and (4,-4) is y = -1x+0 which simplifies to y = -x
The perpendicular line is the one we want. This line will go through (4,-4)
The original slope is m = -1 = -1/1
The perpendicular slope is -1/m = -1/(-1/1) = 1
Recompute the y intercept
y = mx+b
-4 = 1*4+b
-4 = 4+b
-4-4 = 4+b-4
-8 = b
b = -8
The slope intercept equation we want is
y = x-8
Convert to standard form
y = x-8
y-y = x-8-y
0 = x-y-8
x-y-8 = 0
Which is why the answer is choice C
The line x-y-8=0 contains the point (4,-4) which is the closest point to the origin.