Implicit equation of a rotated ellipse

Writing:

$$\left(x-\left(3+\frac{y}{3}\right)t\right)^{2}+y^{2}=1$$

in this way:

$$\left(x-\frac{t}{3}y-3t\right)^{2}+y^{2}=1,$$

you get an ellipse (whatever $t$) as the image of unit circle

$$X^2+Y^2=1$$

by affine change of coordinates:

$$\binom{X}{Y}=\begin{pmatrix}1&-t/3\\0&1\end{pmatrix}\binom{x}{y}+\binom{-3t}{0}$$

This change of coordinates must be written in the inverse way (see alias/alibi transformations here or here) in order to be interpreted as the (affine) transform applied to the unit circle that generates the ellipse with parameter $t$:

$$\binom{x}{y}=\begin{pmatrix}1&t/3\\0&1\end{pmatrix}\binom{X}{Y}+\binom{3t}{0}$$

explaining the combined effect of a distortion of the unit circle (brought by the triangular matrix) followed by an horizontal translation.

Remark: this distorsion is called a shear in the direction of $x$ axis, or sometimes a transvection.