Answer
(a) $2x+3y+12z=24$
(b) $\dfrac{(x-3)}{2}=\dfrac{(y-2)}{3}=\dfrac{(z-1)}{12}$
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$(3,2,1)$
$(x-3)(2)+(y-2)(3)+(z-1)(12)=0$
$2x-6+3y-6+12z-12=0$
This implies, $2x+3y+12z=24$
(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$(3,2,1)$
$\dfrac{(x-3)}{2}=\dfrac{(y-2)}{3}=\dfrac{(z-1)}{12}$
