Maybe your problem comes from the fact that is not the equation of a sphere
Find parametric equations of the line that passes through the center of the sphere
x ^2+ y - 14 y^2 + 53 + z^2 + 4 z = 1
and is perpendicular to the plane
10 x - 6 y + 7 z = -1
The freaking computer keeps telling me that my normal vector and equations of the line are incorrect. Here is what i currently have.