anonymous
  • anonymous
Prove that the area bounded by a tangent at P(cp, c/p) to the rectangular hyperbola xy = c² and the asymptotes of the curve is a constant (ie, not dependent on p) Help please?
Mathematics
jamiebookeater
  • jamiebookeater
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anonymous
  • anonymous
Asymptotes are the x and y axis, and the tangent at P is x + p²y = 8p Other than that, I don't know how to do it
myininaya
  • myininaya
this sounds like you may need integration? do you know integration?
anonymous
  • anonymous
Yep. I just don't know how to apply it to this question

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anonymous
  • anonymous
as x approaches infinity, the hyperbola approaches 0, but it doesn't touch, so I'm not sure what I should integrate since I only have the lower limit
myininaya
  • myininaya
well that tangent line touches the curve sometime after x=0 it touches the tangent at x=cp so assume cp>0 and use as upper limit integrate tanget line - y=c^2/x from 0 to cp and see what happens
anonymous
  • anonymous
oh! wait hold on a sec lemme draw something
anonymous
  • anonymous
1 Attachment
anonymous
  • anonymous
am I trying to find the yellow bit? I was trying to work out the area of the green bit and... yeah, it didn't work
myininaya
  • myininaya
oh thanks for visual i like visuals
anonymous
  • anonymous
lol :)
myininaya
  • myininaya
you need to find the x-intercept of the tangent line
anonymous
  • anonymous
alright, I think I know how to do this now. Thanks!
myininaya
  • myininaya
i will do it too and we can compare answers k :)
anonymous
  • anonymous
okay :) i'll come back when I'm done
anonymous
  • anonymous
oh... i'm using x + p²y = 8p as the equation of the tangent... that's not right is it... I just realised that I got that from a question where c² = 16 so do I have to work out a general one?
myininaya
  • myininaya
i didnt check this but you said to find tangent at (cp,c/p) right ?
1 Attachment
myininaya
  • myininaya
ok i think this is right lol
myininaya
  • myininaya
are you looking at what i did rain?
anonymous
  • anonymous
yep, trying to figure it out
anonymous
  • anonymous
I integrated the wrong thing. I put the equation of the hyperbola instead
myininaya
  • myininaya
right according to your pic we are just wanting the area under that tangent line
myininaya
  • myininaya
from x=0 to x=2cp
anonymous
  • anonymous
y=c^2 /x =c^2 x^(-1) slope m= tangent line=-c^2 x^(-2)= -c^2 /x^2 the slope at P(cp, c/p) m=-c^2 / c^2p^2=1/p^2
anonymous
  • anonymous
ah, okay, I got the answer :) Awesome, thanks so much!!
myininaya
  • myininaya
you got 2c^2?
anonymous
  • anonymous
I haven't finished yet, on the third last line. it's just algebra, so it should be right
anonymous
  • anonymous
okay, got the answer :D Thank you!!!
myininaya
  • myininaya
on that attachemnt i started my work in the second colum n and finished upp inthe first
myininaya
  • myininaya
ok cool rain :)
anonymous
  • anonymous
since m=slope=-c^2/x^2 you can get the EQ of the line y-y1=m(x-x1) y- c/p = (-c^2/ x^2)(x-cp) y = -c^2/x +c^3 p/x^2 +c/p you can now do the area by integration here
anonymous
  • anonymous
good luck Rain, hope you get the correct answer

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