Question

The distance d between two points (x, y) and (a, b) is given by
d =( (x - a)^2 + (y - b)^2)^0.5 .
Use the method of Lagrange Multipliers to find the points on the circle
x^2 - 2x + y^2 - 4y =0 which are nearest to and furthest from the point (-2,-1).
(Hint: Consider d2 subject to the constraint.)

Answer 1

i think lagrange better solve this one

Answer 2

oh he's offline

Answer 3

sorry dont know this

Answer 4

Anybody can help me?!

Answer 5

i solved it but i didn't use Lagrange Multipliers. solve circle equation for y
\[y = 2 \pm \sqrt{5-(x-1)^{2}}\]
Then use distance formula to obtain function in terms of x representing the distance from a point on circle to point (-2,-1)
\[d(x) = \sqrt{(x+2)^{2} +[3+\sqrt{5-(x-1)^{2}}]^{2}}\]
Differentiate and set equal to zero
\[d'(x) = \frac{3+3\frac{1-x}{\sqrt{5-(x-1)^{2}}}}{d(x)} = 0\]
\[\rightarrow x = 1 \pm \sqrt{\frac{5}{2}}\]
\[\rightarrow y = 2 \pm \sqrt{\frac{5}{2}}\]
nearest point (-.58, .419)...........farthest point (2.58,3.58)