Center of curvature and osculating circle

If, along the normal from a point of a curve, one lays off in the direction of concavity of the curve a length equal to the radius of curvature, one obtains the center of curvature. Prove that its coordinates for the curve $x=\varphi (t)$, $y=\psi (t)$ are given by the formulas:

Find the coordinates of the center of curvature of the hyperbola $xy=a^{2}$ at the point $(a,a)$.

Find the coordinates of the center of curvature of the strophoid $y^{2}=x^{2}\dfrac {a+x}{a-x}$.

Find the coordinates of the center of curvature of the curve $y=x\ln x$ at the point where $y'=0$.

Find the coordinates of the center of curvature of the curve $r=a(1+\cos \varphi )$.

Find the coordinates of the center of curvature of the curve $r=a\cos^{3}\varphi$.

Prove that under the same conditions (see the preceding problem) the center of curvature moves along a curve that tends to coincide with the circle $r=a$.

Prove that on an ellipse there exist, in general, three points such that the circles of curvature of the ellipse at these points pass through a given point of the ellipse.

Prove that the centers of curvature at points of the Archimedean spiral $r=a\varphi$ lying on one ray lie on an ellipse whose dimensions do not depend on the choice of the ray.

Find the smallest distance from the center of curvature of an ellipse to the center of the ellipse.

Find the locus of the foci of parabolas having, at a given point, contact of the second order with a given curve.

Find the locus of the midpoints of the chords common to the ellipse $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}=1$ and its osculating circle.

Find the locus of the feet of the perpendiculars dropped from the center of the ellipse $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}=1$ onto the chords common to the ellipse and its osculating circle.