Welcome to our community

Be a part of something great, join today!

Problem Of The Week #410 Mar 27th, 2020

Status
Not open for further replies.
  • Thread starter
  • Admin
  • #1

anemone

MHB POTW Director
Staff member
Feb 14, 2012
3,894
Here is this week's POTW:

-----

Find the minimum value of $(u-v)^2+\left(\sqrt{2-u^2}-\dfrac{9}{v}\right)^2$ for $0<u<\sqrt{2}$ and $v>0$.

-----

Remember to read the POTW submission guidelines to find out how to submit your answers!
 
  • Thread starter
  • Admin
  • #2

anemone

MHB POTW Director
Staff member
Feb 14, 2012
3,894
No one answered last week's POTW.(Sadface)

Below is a suggested solution:
The given function is the square of the distance between a point of the quarter of circle $x^2+y^2=2$ in the open first quadrant and a point of the half hyperbola $xy=9$ in that quadrant. The tangents to the curves at (1, 1) and (3, 3) separate the curves, and both are perpendicular to $x=y$, so those points are at the minimum distance, hence the answer is $(3-1)^2+(3-1)^2=8$.
 
Status
Not open for further replies.