Question

Find the slope of the tangent line to the curve of intersection of the
surface 36𝑥² − 9𝑦² + 4𝑧² + 36 = 0
with the plane 𝑥 = 1 at the point (1, √12, −3) .

Answer 1

at x=1, we have
36 - 9y^2 + 4z^2 + 36 = 0
y^2/8 - z^2/18 = 1
If by slope you mean dy/dz, then
y/4 dy/dz - z/9 = 0
√3/2 dy/dz = -1/3
dy/dz = -2/√27