Lagrange Multipliers
By Light Of Orion on Saturday, May 04,
2002 - 10:20 am:
Hello,
I was working on this problem using Lagrange multipliers.
Find the points in the hyperbola x2 -y2 =1
which is closest to the point (4,0).
I worked through to lambda =(x-4)/2xf=y/(-2yf), where f is the
function of distance between a point in the hyperbola and
(4,0).
Then, 2fy=fxy
fxy-2fy=0
fy(x-2)=0
f=0, y=0, x=2
Now, i understand that x=2 is the solution, but what does that
y=0 and f=0 mean, maybe geometrically?
Thank you.
Regards.
By Ian Short on Saturday, May 04, 2002 -
11:37 am:
Hi, your working is slightly wrong here. If f=0 really
was the distance between the point and hyperbola then the point would have to
lie on the hyperbola. Maybe this is why you asked about the geometry?
The Lagrange multiplier method would consist of minimising (x-4)2 + y2
subject to x2 -y2 = 1. That is, set,
F = (x-4)2 + y2 + l(x2 -y2 -1)
and differentiate with respect to x, y and l.
Fx = 2(x-4) + 2lx = 0 Fy = 2y - 2ly = 0 Fl = x2 -y2 -1 = 0
has solutions with x=1 and y=0 BUT it also, like you say has solution
with l = 1 (x=2 and y=Ö3). This is the minimal solution, as
you observed.
For an alternate geometric interpretation, rotate your axis so that they
coincide with the original asymptotes, explicitly put X=(x-y)/Ö2 and
Y=(x+y)/Ö2 so that the equation is X Y=1/2. Draw a picture- the
simpler equation means finding the minimum distance will be easier by
differentiation.
Ian
By Light Of Orion on Saturday, May 04,
2002 - 07:24 pm:
HI!
Then, what does x=1 and y=0 mean? They are not extreme points i
guess?
By Ian Short on Sunday, May 05, 2002 -
08:42 am:
Hi- I think (1,0) is a local maximum.
Obviously there is no overall maximum value for the distance. (2
, +/- 31/2 ) are unique minima as any minimum point
would have to solve the 3 partial differential equations I
listed.
Try drawing a 3D graph with the hyperbola on the floor in the
xy-plane and the distance from the point to the hyperbola plotted
with the z-axis.
Ian
By Light Of Orion on Sunday, May 05, 2002
- 10:59 am:
Yeah, got it. Thanks a lot Ian.