The Question is :
Find an equation of the plane tangent to the following surface at the given point.
8xy + 5yz + xz - 56 = 0; (2, 2, 2 )
The equation of the tangent plane at (2, 2, 2 ) is 0.
Answer:
The equation of the plane tangent to the surface
8xy + 5yz + xz - 56 = 0
at the point (2, 2, 2 )
is 9x + 13y + 6z - 56 = 0
Step-by-step explanation:
Given the equation
8xy + 5yz + xz - 56 = 0
and the point
(2, 2, 2 ).
To find the equation of the plane tangent to the surface, we first differentiate the given function with respect to x, y and z respectively.
F_x = 8y + z
F_y = 8x + 5z
F_z = 5y + x
At the point (2, 2, 2)
F_x = 18
F_y = 26
F_z = 12
The equation of the plane is given as
(F_x)(x - 2) + (F_y)(y - 2) + (F_z)(z - 2) = 0
18(x - 2) + 26(y - 2) + 12(z - 2) = 0
18x + 26y + 12z - 36 - 52 - 24 = 0
18x + 26y + 12z - 112 = 0
Divide through with 2
9x + 13y + 6z - 56 = 0
This is the equation we are looking for.
