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# rectangular hyperbola question watch

1. you have xy = c^2

and the normal at P(ct, c/t) is

y = t^2 x + c/t - ct^3

so the normal meets the hyperbola again at Q (-c/t^3, -ct^3)

and how do you show that, if the midpoint of PQ is (X/Y) and that t not equal to plus or minus 1,

X/Y = -1/t^2

???
2. To find the midpoint of PQ (X,Y):

X = [ct - c/t^3]/2 = c(t^4-1)/2t^3
Y = [c/t - ct^3]/2 = c(1-t^4)/t = -c(t^4 -1)/2t

=> X/Y = -1/t^2
3. X = (ct - c/t^3)/2 = c(t^4-1)/2t^3
and Y = (c/t - ct^3)/2 = c(1-t^4)t^3/2t
So X/Y = -1/t^2
4. that was easy
5. I spoke to me da about it and he says that it's not the one x value minus the other divided by two. He says you have to add the two values together and then divide them by two. The problem is, how did he get the right answer and you get the right answer using one right, one wrong, method?
But the point Q has a negative value for x and a negative value for y, hence the midpoint of PQ is:

Xmid = [-c/t^3 + ct]/2 = [ct - c/t^3]/2

and same for Ymid...

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