Parametric circle and ellipse with points

(a) Circle

$$x=x_c+R\cos\theta,\qquad y=y_c+R\sin\theta,\qquad 0\le\theta\le2\pi$$

With $(x_c,y_c)=(2,3)$, $R=5$: $x=2+5\cos\theta$, $y=3+5\sin\theta$.

Tangent: $\dfrac{dx}{d\theta}=-5\sin\theta,\ \dfrac{dy}{d\theta}=5\cos\theta$. At $60^\circ$: $(-4.330,\ 2.500)$. Its length is 5, and it is perpendicular to the radius vector $(2.5,\ 4.330)$, since the dot product is $-10.83+10.83=0$.

(b) Ellipse

$$x=x_c+a\cos\theta,\qquad y=y_c+b\sin\theta$$

With $a=6$, $b=3$, centre $(1,2)$: $x=1+6\cos\theta$, $y=2+3\sin\theta$.

Check at 45 deg: $\dfrac{(5.243-1)^2}{36}+\dfrac{(4.121-2)^2}{9}=0.5+0.5=1$.

The circle is the special case $a=b=R$. Note that $\theta$ here is a parameter (the eccentric angle), not the polar angle of the point on the ellipse.

Answer: circle points (7, 3), (4.5, 7.33), (-0.5, 7.33), (-3, 3); tangent at 60 deg = (-4.33, 2.5); ellipse points (7, 2), (5.243, 4.121), (1, 5).