Write $\mathbb{R}^N=\{(x_1,x_2,...,x_N)|\ x_j \in \mathbb{R} \ \forall \ j=1,...,N\}$ and consider a point $a$ in the hyperplane $x_1=1$. Does there exist a hyper-ellipsoid $E_a$, with non-zero volume in $\mathbb{R}^N$ and centered at the origin such that the hyperplane $x_1=1$ is tangent to $E_a$ at $a$?
Work so far:
This is true for $N=2$, since, in $\mathbb{R}^2$, $x^2+bxy+cy^2=1$ is the equation for an ellipse provided $b^2 \leq 4ac$. For the point $a=(1,y_0)$, with $y_0 \neq 0$, the ellipse
$2x^2 - \dfrac{2xy}{y_0} + \dfrac{y^2}{y_0^2}=1$
is centered at the origin and is tangent to the line $x=1$ at $y_0$. If $y_0=0$, the unit circle works. Does this property generalise for $N>2$?
Extending your example, the ellipsoid $$2x_1^2 - \frac{2x_1x_2}{t} + \frac{x_2^2}{t^2} + x_3^2 + x_4^2 + \dots + x_n^2 = 1$$ is centered at the origin and tangent to the hyperplane $x_1 = 1$ at $P = (1,t,0,0,\dots,0,0)$. We may then apply a rotation to the last $n-1$ coordinates which will preserve the hyperplane but move the point $P$ to an arbitrary point that's a distance $t$ away from $(1,0,0,\dots,0,0)$.
So yes, the property generalizes.