Since $\displaystyle \frac{x}{a}$ accomplishes a horizontal shear of magnitude $a$ and similarly for $\displaystyle \frac{y}{b}$ in the vertical direction, the unit circle $x^2 + y^2 =1$ after the shear will be the ellipse:
$\left(\frac{x}{a}\right)^{2} + \left(\frac{y}{b}\right)^{2} = 1$
In addition, if the semi-major axis is allowed to remain length $1$, $a=1$ and $0. The latter suggests $b$ can be represented by $\sin(\theta)$. Explore and you will find that this unit ellipse has foci at $\pm \cos(\theta)$, directrices at $\pm \sec(\theta)$ and focal width $2\sin^{2}(\theta)$.
The connections between the unit circle, transformations and trigonometry are important for secondary students to understand. It is somewhat typical of textbooks to introduce a topic like conic sections with formulas of their own as if they were a different species of mathematical functions.
Even our new Common Core does not use the approach. If you Google "circle transformation ellipse" you will see that Archimedes noted this connection.