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June 8th, 2014, 11:25 AM   #1
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deriving a line's equation

I am attempting to find the equation of a 'perpendicular bisector'. So let's assume I have calculated the gradient and a point on the line correctly, they are:

(-1,1) with a gradient of -5

I now ought to be able to find the equation of the line [ax + by +c = 0]:

y - y1 = m(x - x1)

y - 1 = -5 (x + 1)

5y + 5 = -5x -5

5y = - 5x

5x + 5y = 0
------------------------

However, apparently the right answer is x + 5y -4 so either my working are wrong or calculations to work out the properties of the 'perpendicular bisector' are wrong. [(-1,1) with a gradient of -5]

any thoughts?
Mark
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June 8th, 2014, 11:46 AM   #2
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Can you post the entire problem, as given?
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June 8th, 2014, 01:08 PM   #3
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"A perpendicular bisector is a line which cuts another line in half, at right angles to the first line. Find the perpendicular bisector of the line PQ, where P = (-2,-4) and Q = (0,6). Arrange the equation of your line in the form ax + by + c = 0."

So we have the mid point of PQ and can work out the gradient of the bisector line, and from there produce the line's equation. Sounds so easy...

cheers,
Mark
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June 8th, 2014, 03:41 PM   #4
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if the line goes from the point P(-2,-4) to Q(0,6) then the gradient is calculated as the change in y over the change in x, which is 10/2=5

you have found the midpoint correctly which is (-1,1)

the gradient of a perpendicular to a line of y=mx+c is -1/m which in this case is -1/5 = -0.2

we can now write out the equation for the perp and then fill in what we know:
y=mx+c (now sub in our -0.2 for gradient of perpendicular)
y=-0.2x+c (now sub in the values of the midpoint for x and y)
1=-0.2(-1) + c (now solve to find c)
1=0.2+c
c=0.8

hence equation of perpendicular is y=-0.2x + 0.8
however your question wants only whole numbers, hence if we multiply by 5 we have 5y = -x + 4 then add x to both sides
5y + x = 4 then subtract 4 from both sides gives
5y + x - 4 = 0

hope that kind of explains it, need anything explained better or anything that's not clear let me know
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June 9th, 2014, 12:33 PM   #5
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Superb!

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