Which of the following equations describe a tangent plane to the curve z=1/xy, at the point (x,y)=(1,2).

Question

Which of the following equations describe a tangent plane to the curve z=1/ xy, at the point (x,y)=(1,2).

a)z - 1/2 = -1/2(x-1) - 1/4(y-2)

b)z - 1/2 = 1/2(x-1) + 1/4(y-2)

c) z - 1/2 = 1/4(x-1) + 1/2(y-2)

d)z - 1/2 = -1/4(x-1) - 1/2(y-2)

Answer 1

z=1/xy or

xyz = 1

xydz+ xzdy + yzdx =0

(x, y) =(1,2), so z = 1/xy = 1/(1*2) =1/2

1*2dz + 1*1/2dy + 2*1/2dx =0

2dz + 1/2dy + 1dx =0

2z +1/2y + x = C

2z +1/2y + x = 2(1/2)+1/2(2) + 1(1)= 3

2z +1/2y + x = 3,or

4z + y + 2x = 6