The given ellipsoid and sphere, respectively, are $$\psi(x, y, z)=2x^2+10y^2+z^2-2=0,$$ $$\xi(x, y, z)=x^2+y^2+z^2-x-4y-2z+8=0.$$Given that above two are tangent to each other at the point $P=(r, s, t)$ where $r=\frac{5}{8}.$ Find $s$ and $t$.
My attempt: I calculated the tangent planes of $\psi$ and $\xi$ and find them at $(r, s, t)$, respectively, as $$4r(x-r)+20s(y-s)+2t(z-t)=0,$$ $$(2r-1)(x-r)+(2s-4)(y-s)+(2t-2)(z-t)=0.$$ Now if the ellipsoid and the sphere are tangent to each other at a point, then they must have the same tangent plane at $P$, i.e. we must have $$\frac{2r-1}{4r}=\frac{2s-4}{20s}=\frac{2t-2}{2t}=z\ \text{where z is some constant}.$$ But after putting $r=\frac58$, I get $z=\frac{1}{10}$. Due to this, I am unable to deduce $s$. Please tell where I am wrong.