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We wish to min (x-0)^2 + (y-14)^2

subject to (x-2)^2 + y^2 - 2 = 0

We form L(x,y) = x^2 + (y-14)^2 - k( (x-2)^2 + y^2 - 2 )

Diff wrt k Lk = (x-2)^2 + y^2 - 2 = 0

Diff wrt x Lx = 2x - 2k(x-2) = 0 => k = x/(x-2)

Diff wrt y Ly = 2(y-14) - 2ky = 0 => k = y/(y-14)

From last 2 eqns we get x/(x-2) = y/(y-14)

xy = (x-2)(y-14) = xy - 14x - 2y + 28

so 7x + y = 14 => y = 7(2-x) or y^2 = 49(x-2)^2

From the eqn for Lk we now get (x-2)^2 + 49(x-2)^2 = 2

or (x-2)^2 = 2/50

x-2 = +-1/5

x = 2+-1/5 or x = 9/5 , 11/5

We also have y^2 = 49(x-2)^2 => y^2 = 7/25

y = +-sqrt(7)/5

So we have 4 pts correspind to x = 9/5, 11/5 and y=+-sqrt(7)/5

Evaluate the x^2 + (y-14)^2 at these 4 points to see which gives the min

subject to (x-2)^2 + y^2 - 2 = 0

We form L(x,y) = x^2 + (y-14)^2 - k( (x-2)^2 + y^2 - 2 )

Diff wrt k Lk = (x-2)^2 + y^2 - 2 = 0

Diff wrt x Lx = 2x - 2k(x-2) = 0 => k = x/(x-2)

Diff wrt y Ly = 2(y-14) - 2ky = 0 => k = y/(y-14)

From last 2 eqns we get x/(x-2) = y/(y-14)

xy = (x-2)(y-14) = xy - 14x - 2y + 28

so 7x + y = 14 => y = 7(2-x) or y^2 = 49(x-2)^2

From the eqn for Lk we now get (x-2)^2 + 49(x-2)^2 = 2

or (x-2)^2 = 2/50

x-2 = +-1/5

x = 2+-1/5 or x = 9/5 , 11/5

We also have y^2 = 49(x-2)^2 => y^2 = 7/25

y = +-sqrt(7)/5

So we have 4 pts correspind to x = 9/5, 11/5 and y=+-sqrt(7)/5

Evaluate the x^2 + (y-14)^2 at these 4 points to see which gives the min

Indikos
· 4 months ago

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