Suppose I am given the curve $f(x,y)=\frac{x^2}{p^2}+\frac{y^2}{q^2}=1$. I want to write down explicitly an equation that parametrizes all tangent lines to this ellipse. In this question, the answers freely use the fact that the equation of any tangent line at a point $(x_0,y_0)$ of the ellipse can be written as $$\frac{x}{p^2}x_0+\frac{y}{q^2}y_0=1\space\text{or}\space\frac{x}{p^2}\left(p\cos(t)\right)+\frac{y}{q^2}q\sin(t)=1$$
Why is this?
Assume that $(x_0,y_0)$ is a point of the ellipse with $y_0\ne 0.$ Deriving with respect to $x$ in $$\frac{x^2}{p^2}+\frac{y^2}{q^2}=1$$ we get
$$\frac{x}{p^2}+\frac{yy'}{q^2}=0$$ from where
$$y'=-\frac{q^2x}{yp^2}.$$ The equation of the tangent line is
$$y-y_0=y'(x_0)(x-x_0).$$ That is,
$$y-y_0=-\frac{q^2x_0}{y_0p^2}(x-x_0).$$ In other words
$$\frac{x}{p^2}x_0+\frac{y}{q^2}y_0=1.$$
If $y_0=0$ you can proceed in a similar way taking derivatives with respect to $y.$
