anonymous
  • anonymous
Help!!! Find the differential equation of all the circles with centers x + 2y = 0 and passing through (0,0).
Mathematics
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anonymous
  • anonymous
Help!!! Find the differential equation of all the circles with centers x + 2y = 0 and passing through (0,0).
Mathematics
chestercat
  • chestercat
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anonymous
  • anonymous
How about this... A circle with center (-2u, u), radius r is written as \[(x + 2u)^{2} + (y - u)^{2} = r^{2}\] Differentiating both side with respect to x, we get \[ x + 2u + (y - u)y' = 0\]
anonymous
  • anonymous
how do you get the center?
anonymous
  • anonymous
This circle \[ (x+2u)^2+(y−u^)2=r^2\] has its centre at (-2u, u), right?

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anonymous
  • anonymous
the centers of the all circles are lying at a line x + 2y = 0. how do you get the center (-2u,u) and the circle must passing through (0,0).
anonymous
  • anonymous
Let x = -2u and y = u. For any u, (-2u) + 2u = 0. So (-2u, u) is on the line x + 2y = 0. I got this pair (-2u, u) just by modifying x + 2y = 0 -> x = -2y.
anonymous
  • anonymous
ok, thanks
anonymous
  • anonymous
I'm glad if it helped. Thanks.

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