Parametrizing plane curves by an auxiliary parameter

Using the identity $\cos^{2}t+\sin^{2}t=1$, represent the equation of the ellipse $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}=1$ in parametric form.

Using the identity $e^{-t}\cdot e^{t}=1$, represent the equation of the hyperbola $\dfrac {x^{2}}{a^{2}}-\dfrac {y^{2}}{b^{2}}=1$ in parametric form.

Represent the equation of the circle $x^{2}+y^{2}=a^{2}$ in parametric form, taking as parameter the slope of a chord of the circle passing through the point $(-a, 0)$.

Prove that the coordinates of the points of a second-order curve can be expressed rationally in terms of the slope of a variable chord of the curve passing through some chosen point of the curve.

A curve $F(x,y)=0$ whose points' coordinates can be expressed rationally in terms of a parameter is called unicursal. Prove that the curves $r = f(\cos \varphi , \sin \varphi )$, where $f$ is a rational function, are unicursal.

Prove that the cissoid $y^{2} = \dfrac {x^{3}}{2a-x}$ is unicursal, by expressing the coordinates of its points in terms of the slope of the chord joining the origin to a given point of the curve.

Solving simultaneously the equation of the folium of Descartes $x^{3}+y^{3}=3axy$ and the equation of the auxiliary line $y = tx$, obtain the parametric equations of the folium of Descartes:

Prove that three distinct points of the folium of Descartes, corresponding to the values $t_{1}$, $t_{2}$, and $t_{3}$ of the parameter, lie on one line if and only if $t_{1}t_{2}t_{3} = -1$.

Prove that the coordinates of the points of the cardioid $r = a(1+\cos \varphi )$ can be expressed in terms of some parameter $t$ by the formulas:

Under what condition do three points of the cardioid, corresponding to the values $t_{1}$, $t_{2}$, $t_{3}$ of the parameter (see the previous problem), lie on one line?

Prove that the coordinates of the points of the strophoid $(2a-x)y^{2} = x(x-a)^{2}$ can be expressed rationally in terms of some parameter $t$ by the formulas:

Under what condition do the points of the strophoid corresponding to the values $t_{1}$, $t_{2}$, $t_{3}$ of the parameter (see the previous problem) lie on one line?

For what values of $t_{1}$, $t_{2}$, $t_{3}$, $t_{4}$ do the corresponding points of the strophoid (see problem 1909) lie on one circle?

Prove that the four points of the cissoid

corresponding to the values $t_{1}$, $t_{2}$, $t_{3}$, and $t_{4}$ of the parameter, lie on one circle provided that $t_{1}+t_{2}+t_{3}+t_{4}=0$.

Express rationally, in terms of an auxiliary parameter, the coordinates of the points of the lemniscate $(x^{2}+y^{2})^{2}=a^{2}(x^{2}-y^{2})$.

Show that the curve $x^{y}=y^{x}$ splits into the bisector of the first coordinate angle and the curve

where $n$ is a variable parameter.