Lagrange multipliers to find the points

Question

Use the method of Lagrange multipliers to find the points on the circle x² + y² = 52 that are closest to and farthest from the point (−3, 2).

Answer 1

f(x,y) = x² + y² - 52 = 0

Lagrange function: L(x, y, λ) = (x + 3)² + (y - 2)² - λ(x² + ,y² - 52)

∂L/∂x = 2(x + 3) - 2λx = 0; ∂L/∂y = 2(y - 2) - 2λy = 0; ∂L/∂λ = - x² - y² + 52 = 0;

x = 3/(λ - 1); y = 2/.(1 - λ); 9/(λ - 1)² + 4/(λ - 1)² = 52; (λ - 1)² = 1/4; λ = 3/2 or λ = 1/2

λ = 1/2, then x = - 6; y = 4 Point (- 6, 4) distance: d = [(- 6 + 3)² + (4 - 2)²]^1/2 = √13 min

λ = 3/2; then x = 6; y = - 4. Point (6, - 4); distance d = [( 6 + 3)² + (- 4 - 2)²]^1/2 = 3√13 max