Question

77 Find the • points on the rotated ellipse xxxyty² = 3 that are newest and are faret away 1 from the origin. (Hint: rote that the septem of linear equations axtby=o has non-trivial equations iff ad-bc=0 1 extdy so

Answer 1

Ans) Assume that a point (1,9) ia on the rotated evipse *?+xg+Y?=3 and its distance from ongin is given by d = 5x²2y2 To minimised, need to minimise d² d² = f(x,y) d² = x² + y² Hence, f(X,Y)= x² + y 2 0 This distance is subject to constraints. g(x) = x² + xy + y 2-3 The partial derivative of these tur are FX(X,Y)=20 9X(X,Y)=2x+y fy(2,y)=ay gy(339) = 24th use lagrange multiplier methods Let f(x,y), 8(9,8) be continuously ditterentiable tulog+ the maximum and minimum values of Poly) subject to constraint 964,9)=0 at the point p cohere Tgp to thon F = d vg By wing the lagrange maxiplier method 914,4)=0 Ex(1, y) = dgn(ay) Fy (21, y) = dgylny) Here des called the Lagrange multiplier. Substitute the values of g(x,y) , fx (2, 4), fy(x,y),gx (4,4) and Gy(4,7). 2x=andtyd Qay= any d+y² 2y = 24 dtud gay Qayd told 1 6 Eq and @ can be written as 212-2d) - Yd=0 And anty(20-2)=0

These Eqn will have non tivial son it 2-24 d 2d-2 That to, - (20-2²+d²=0 -4d²+8d-4ta²=0 30²_8d+y=0 3012_6d-ad+y=0 3d (8-2) -2(0-2) =0 d=212 From Ear and 6 y²d = x²d y=+x From Ea ② x²+x(+1) + ( + x)² =3 222 +2²3 3x23x²=3 >= $1,53 Y = t1, 53 The Reuuting points are, (1,-1),(,1),(-1,2), (,-1), (13,53): (15,-43),(-13,-53),(-53,75) This implies 18,3)=2 at the points (1,-1),0,1),(-1,1),(-1,-1) and f(x,y) = 6 at the points (53,53), (13,-13),(-13, 13),(-13,-15) Therettye the points closest to origin aree (1,-0,6-1,-1), (-1,1),(1,1) and points tou theat tomorigin are (13, 13), (13, -53).(-5,55),(-13,-43)