Order of contact (tangency) of curves

Choose the parameters $k$ and $b$ of the line

so that it has tangency of order higher than one with the curve

How should the coefficients $a, b$ and $c$ be chosen so that the parabola

has second-order tangency with the curve $y=e^{x}$ at the point $x=x_{0}$?

What is the order of tangency of each of the following curves with the $Ox$ axis at the point $x=0$:

Prove that the curve: $y=e^{-\frac{1}{x^{2}}}$ for $x \neq 0$, and $y=0$ for $x=0$, has tangency of infinite order with the $Ox$ axis at the point $x=0$.

Find a parabola joining the origin $O$ to the point $B(a+b,0)$ so that the arc $OB$ of the parabola, together with the lower half of the circle $(x-a)^{2}+y^{2}=b^{2}$, forms a smooth curve with continuous curvature.

Find the parabola $y=ax^{2}+bx+c$ that has, with the sinusoid $y=\sin x$, at the point $\left(\dfrac {\pi }{2}, 1\right)$, a common tangent and a common curvature.

Find the fifth-degree parabola

which at the points $(a,h)$, $(-a,-h)$ is tangent to the lines $y=h$ and $y=-h$, having contact of the second order.

Prove that the circle $(x-3a)^{2}+(y-3a)^{2}=8a^{2}$ and the parabola $\sqrt{x}+\sqrt{y}=2\sqrt{a}$ have contact of the third order at the point $(a,a)$.

Prove that the osculating circle (circle of curvature) of a second-order curve at the vertex of the curve has contact of the third order.

Prove that for every point at which a curve is sufficiently smooth, there exists a second-order curve having, with the given curve at the given point, contact of the fourth order.

Prove that for a cycloid, the second-order curve with contact of the fourth order is always an ellipse.

Prove that for a logarithmic spiral, the second-order curves having contact of the fourth order with it are ellipses whose major axes make a constant angle with the radius vector of the point of tangency.