Step 1 :First, we need to find the gradient vectors of the two given spheres at point P. The gradient of a function F(x, y, z) is given by \(\nabla F = \left(\frac{dF}{dx}, \frac{dF}{dy}, \frac{dF}{dz}\right)\).
Step 2 :For the first sphere, F(x, y, z) = x^2 + y^2 + z^2 - 27, we have: \(\nabla F_P = \left(\frac{dF}{dx}, \frac{dF}{dy}, \frac{dF}{dz}\right) = (2x, 2y, 2z)\) evaluated at P = (1, 1, 5) gives \(\nabla F_P = (2, 2, 10)\).
Step 3 :For the second sphere, G(x, y, z) = (x-2)^2 + (y-2)^2 + z^2 - 27, we have: \(\nabla G_P = \left(\frac{dG}{dx}, \frac{dG}{dy}, \frac{dG}{dz}\right) = (2(x-2), 2(y-2), 2z)\) evaluated at P = (1, 1, 5) gives \(\nabla G_P = (-2, -2, 10)\).
Step 4 :Next, we find the cross product of these two gradient vectors, which will give us the direction vector of the tangent line to the curve C at point P. \(\nabla F_P \times \nabla G_P = (2, 2, 10) \times (-2, -2, 10) = (0, 0, 0)\).
Step 5 :This result indicates that the two gradient vectors are parallel, which means the two spheres are tangent at point P. Therefore, the tangent line to the curve C at point P does not exist. \(\boxed{\text{The tangent line does not exist}}\)