Osculating plane of a space curve

Find the osculating plane of the curve

at the point $(1,1,1)$.

Find the osculating plane of the curve

Find the osculating plane of the curve

Find the osculating plane of the curve $x=e^{t}$, $y=e^{-t}$, $z=t\sqrt{2}$.

Find the osculating plane of the curve $x=a\cos t$, $y=a\sin t$, $z=bt$.

Find the osculating plane of the curve $x=at^{2}+bt+c$, $y=a_{1}t^{2}+b_{1}t+c_{1}$, $z=a_{2}t^{2}+b_{2}t+c_{2}$.

Prove that the osculating plane of the curve

lying on the conoidal surface $z=\varphi \left(\dfrac {y}{x}\right)$, coincides with the tangent plane to this surface.

Write the equations of the osculating, normal, and rectifying planes at an arbitrary point of the curve:

Prove that the line drawn from an arbitrary point $M$ of the curve $x = t, y = t^{2}, z = t^{3}$ parallel to the plane $z = 0$ until it meets the $z$-axis lies in the osculating plane of the curve at the point $M$.

Write the equation of the osculating plane of the curve $x = a \cos t, y = b \sin t, z = e^{t}$ at $t = 0$.

Write the equation of the osculating plane at the point $(2 ; 1 ; 2)$ of the curve that is the intersection of the sphere $x^{2}+y^{2}+z^{2} = 9$ and the hyperbolic cylinder $x^{2}-y^{2} = 3$.

Let $\Gamma = \left\{ M(t) ; a \leqslant t \leqslant b\right\}$ be a twice continuously differentiable curve without singular points, $t_{0} \in [a ; b], t_{0}+\Delta t_{1} \in [a ; b]$ and $t_{0} + \Delta t_{2} \in [a ; b]$. Draw a plane through the points $M\left(t_{0}\right), M\left(t_{0}+\Delta t_{1}\right)$ and $M\left(t_{0}+\Delta t_{2}\right)$. Prove that if at the point $M\left(t_{0}\right)$ the curvature is nonzero, then as $\Delta t_{1} \rightarrow 0$ and $\Delta t_{2} \rightarrow 0$ this plane tends (define this notion) to the osculating plane at the point $M\left(t_{0}\right)$.

Let $\pi$ be the plane passing through the tangent line at the point $M\left(s_{0}\right)$ of a twice continuously differentiable curve $\Gamma = \left\{ M(s) ; 0 \leqslant s \leqslant S\right\}$, where $s$ is the variable arc length of the curve $\Gamma , s_{0} \in [0, S]$, and let $d(\Delta s)$ be the distance from the point $M\left(s_{0}+\Delta s\right)$ to the plane $\pi$.

Prove that the plane $\pi$ is the osculating plane of the curve $\Gamma$ if and only if