Is the Euclidean distance monotonic under affine transformation?

Question

Let us say we have an Affine transformation: $\vec{\gamma}=\eta \vec{\tau} +\vec{\kappa}$, where $\eta$ is a diagonal matrix. Specifically, $\vec{\tau}{=}[\tau_x \ \tau_y\ \tau_z]^T$ forms a sphere, i.e. $\tau_x^2 + \tau_y^2 + \tau_z^2 = 1$. The resulting affine transformation distorts the sphere into an ellipsoid: $\frac{(\gamma_x-\kappa_x)}{\eta_x}^2 + \frac{(\gamma_y-\kappa_y)}{\eta_y}^2 + \frac{(\gamma_z-\kappa_z)}{\eta_z}^2 = 1$. The matrix $\eta$ and the vector $\vec{\kappa}$ are such that the resulting ellipsoid lies within the initial sphere. Under these conditions, if the two points on the ellipsoid, $\vec{\gamma_1}$ and $\vec{\gamma_2}$, have the highest Euclidean distance, then can we say that the corresponding points $\vec{\tau_1}$ and $\vec{\tau_2}$ on the sphere have the highest distance, i.e., do they lie on the opposite ends on the sphere?