Answer
The two surfaces are tangent to each other at the point $(1,1,2)$. This means that they have a common tangent plane at the point.
Work Step by Step
Formula to calculate tangent plane equation for an ellipsoid is:
$(x_2-x_1)f_x(x_1,y_1,z_1)+(y_2-y_1)f_y(x_1,y_1,z_1)+(z_2-z_1)f_x(x_1,y_1,z_1)=0$
$(x-x_0)(6x)+(y-y_0)(4y)+(z-z_0)(-2z)=0$
At point$(1,1,2)$
$(x-1)(6)+(y-1)(4)+(z+1)(-4)=0$
$6x-6+4y-4-4z-4=0$
$6x+4y-4z=18$ ....(1)
Formula to calculate tangent plane equation for an sphere is:
$(x_2-x_1)f_x(x_1,y_1,z_1)+(y_2-y_1)f_y(x_1,y_1,z_1)+(z_2-z_1)f_x(x_1,y_1,z_1)=0$
$(x-x_0)(2x-8)+(y-y_0)(2y-6)+(z-z_0)(2z-8)=0$
At point$(1,1,2)$
$(x-1)(6)+(y-1)(4)+(z+1)(-4)=0$
$6x-6+4y-4-4z-4=0$
$6x+4y-4z=18$ ...(2)
From equations (1) and (2), we get that the two surfaces are tangent to each other at the point $(1,1,2)$. This means that they have a common tangent plane at the point.
