Respuesta :
Answer:
[tex]8x+8y-z+17 = 0[/tex]
Step-by-step explanation:
We are given the following information in the question:
We are given a surface
[tex]z = 4(x-1)^2 + 4(y+3)^2 + 9[/tex]
[tex]F(x,y,z) = z - 4(x-1)^2 - 4(y+3)^2 - 9 = 0[/tex]
We have to find the equation of tangent passing through the point (2, -1, 17).
The equation of tangent is given by:
[tex]F_x(x_0,y_0, z_0).(x-x_0) + F_y(x_0,y_0, z_0).(y-y_0) + F_z(x_0,y_0, z_0).(z-z_0) = 0[/tex]
Evaluating:
[tex]F_x = -8(x-1)\\F_x(2,-2,17) = -8(2-1) = -8\\F_y = -8(y+3)\\F_y(2,-2,17) = -8(-2+3) = -8\\F_z = 1\\F_z(2,-2,17) = 1\\[/tex]
Putting all the values in equation of tangent:
[tex]-8(x-2) -8(y+2) + 1(z-17) = 0\\-8x + 16 -8y - 16 + z -17 = 0\\-8x-8y+z-17 = 0\\8x+8y-z+17 = 0[/tex]
