Answer
(a) $x+y+z=3$
(b) $x=y=z$
Work Step by Step
(a) Formula to calculate tangent plane equation is:
$(x_2-x_1)f_x(x_1,y_1,z_1)+(y_2-y_1)f_y(x_1,y_1,z_1)+(z_2-z_1)f_x(x_1,y_1,z_1)=0$
At point$(1,1,1)$
$(x-1)(-2)+(y-1)(-2)+(z-1)(-2)=0$
This implies, $x+y+z=3$
(b) Formula to normal line equation is:
$\dfrac{(x_2-x_1)}{f_x(x_1,y_1,z_1)}=\dfrac{(y_2-y_1)}{f_y(x_1,y_1,z_1)}=\dfrac{(z_2-z_1)}{f_x(x_1,y_1,z_1)}$
At point$(1,1,1)$
$\dfrac{(x-1)}{-2}=\dfrac{(y-1)}{-2}=\dfrac{(z-1)}{-2}$
Hence, $x=y=z$
