Applications in Geometry

Write the tangent line and normal plane equations at the given point for the following curve: $x=a\cos \alpha \cos t$, $y=a\sin \alpha \cos t$, $z=a\sin t$; at the point $t=t_{0}$.

Write the tangent line and normal plane equations at the given point for the following curve: $x=a\sin^{2}t$, $y=b\sin t\cos t$, $z=c\cos^{2}t$; at the point $t=\dfrac {\pi }{4}$.

Write the tangent line and normal plane equations at the given point for the following curve: $y=x$, $z=x^{2}$; at the point $M(1,1,1)$.

Write the tangent line and normal plane equations at the given point for the following curve: $x^{2}+z^{2}=10$, $y^{2}+z^{2}=10$; at the point $M(1,1,3)$.

Write the tangent line and normal plane equations at the given point for the following curve: $x^{2}+y^{2}+z^{2}=6$, $x+y+z=0$; at the point $M(1,-2,1)$.

Find a point on the curve $x=t$, $y=t^{2}$, $z=t^{3}$ at which the tangent line is parallel to the plane $x+2y+z=4$.

Prove that the tangent line to the helix $x=a\cos t$, $y=a\sin t$, $z=bt$ makes a constant angle with the $Oz$ axis.

Prove that the curve

intersects all the generating lines of the cone $x^{2}+y^{2}=z^{2}$ at the same angle.

Prove that the loxodrome

(where $\varphi$ is the longitude of a point on the Earth, and $\psi$ is the latitude of a point on the Earth) intersects all meridians of the Earth at a constant angle.

Given the curve

where $f$ is a differentiable function. Find the tangent of the angle formed by the tangent line to the curve at the point $M_{0}(x_{0}, y_{0})$ with the $Oxy$ plane.

Find the derivative of the function

at the point $M(1,2,-2)$ along the direction of the tangent line, at this point, to the curve $x=t$, $y=2t^{2}$, $z=-2t^{4}$.

Write the tangent plane and normal line equations at the given point for the following surface: $z=x^{2}+y^{2}$; at the point $M_{0}(1,2,5)$.

Write the tangent plane and normal line equations at the given point for the following surface: $x^{2}+y^{2}+z^{2}=169$; at the point $M_{0}(3,4,12)$.

Write the tangent plane and normal line equations at the given point for the following surface: $z=\operatorname {arctg}\dfrac {y}{x}$; at the point $M_{0}\left(1, 1, \dfrac {\pi }{4}\right)$.

Write the tangent plane and normal line equations at the given point for the following surface: $ax^{2}+by^{2}+cz^{2}=1$; at the point $M_{0}(x_{0}, y_{0}, z_{0})$.

Write the tangent plane and normal line equations at the given point for the following surface: $z=y+\ln \dfrac {x}{z}$; at the point $M_{0}(1,1,1)$.

Write the tangent plane and normal line equations at the given point for the following surface: $2^{\frac{x}{z}}+2^{\frac{y}{z}}=8$; at the point $M_{0}(2,2,1)$.

Write the tangent plane and normal line equations at the given point for the following surface: $x=a\cos \psi \cos \varphi$, $y=b\cos \psi \sin \varphi$, $z=c\sin \psi$; at the point $M_{0}(\varphi_{0}, \psi_{0})$.

Write the tangent plane and normal line equations at the given point for the following surface: $x=r\cos \varphi$, $y=r\sin \varphi$, $z=r\operatorname {ctg}\alpha$; at the point $M_{0}(\varphi_{0}, r_{0})$.

Write the tangent plane and normal line equations at the given point for the following surface: $x=u\cos v$, $y=u\sin v$, $z=av$; at the point $M_{0}(u_{0}, v_{0})$.

Find the limiting position of the tangent plane to the surface

at the point of tangency $M(u,v)$ $(u \neq v)$, as it approaches without bound a point $M_{0}(u_{0}, v_{0})$ on the boundary curve $u=v$ of the surface.

Find on the surface $x^{2}+2y^{2}+3z^{2}+2xy+2xz+4yz=8$ all the points of tangency at which the tangent plane is parallel to a coordinate plane.

At which points on the ellipsoid

does the normal line to the ellipsoid make equal angles with the coordinate axes?

Find the tangent planes to the surface $x^{2}+2y^{2}+3z^{2}=21$ that are parallel to the plane

Prove that the tangent planes to the surface $xyz=a^{3}$ $(a>0)$ form, with the coordinate planes, a tetrahedron of constant volume.

Prove that the sum of the segments cut off on the coordinate axes by the tangent plane to the surface

is a constant.

Prove that the tangent plane of the conical surface

passes through its vertex.

Prove that the normal of the surface of revolution

meets the axis of rotation.

Find the projections of the ellipsoid

onto the coordinate planes.

Divide the square $\left\{ 0 \leqslant x \leqslant 1, 0 \leqslant y \leqslant 1\right\}$ into finitely many parts $\sigma$ of diameter $\leqslant \delta$. If the surface

is such that at any two points $P(x,y)$ and $P_{1}(x_{1}, y_{1})$ belonging to the same part $\sigma$ the normal directions differ by less than $1^{\circ }$, find an upper bound for the number $\delta$.

Let

be the equation of a surface, and let $\varphi (P_{1}, P)$ be the angle between the normals of the surface at the two points $P(x,y) \in D$ and $P_{1}(x_{1}, y_{1}) \in D$.

Prove that if the domain $D$ is bounded and closed, and the function $f(x,y)$ has bounded second-order derivatives in $D$, then the Lyapunov inequality

holds, where $C$ is a constant and $\rho (P_{1}, P)$ is the distance between the points $P$ and $P_{1}$.

At what angle does the cylinder $x^{2}+y^{2}=a^{2}$ intersect the surface $bz=xy$ at a common point $M_{0}(x_{0}, y_{0}, z_{0})$?

Prove that the coordinate surfaces of spherical coordinates $x^{2}+y^{2}+z^{2}=r^{2}$, $y=x\operatorname {tg}\varphi$, $x^{2}+y^{2}=z^{2}\operatorname {tg}^{2}\theta$ are pairwise orthogonal.

Prove that the spheres $x^{2}+y^{2}+z^{2}=2ax$, $x^{2}+y^{2}+z^{2}=2by$, $x^{2}+y^{2}+z^{2}=2cz$ form a triply orthogonal system.

When $\lambda =\lambda_{1}$, $\lambda =\lambda_{2}$, $\lambda =\lambda_{3}$, through each point $M(x,y,z)$ there pass three quadric surfaces:

Prove that these surfaces are orthogonal.

Find the derivative of the function $u=x+y+z$ along the outward normal direction at the point $M_{0}(x_{0}, y_{0}, z_{0})$ on the sphere $x^{2}+y^{2}+z^{2}=1$.

At which points on the sphere does this derivative attain:

Find the derivative of the function $u=x^{2}+y^{2}+z^{2}$ along the outward normal direction at the point $M_{0}(x_{0}, y_{0}, z_{0})$ on the ellipsoid $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}+\dfrac {z^{2}}{c^{2}}=1$.

Let $\dfrac {\partial u}{\partial n}$ and $\dfrac {\partial v}{\partial n}$ be the derivatives of the functions $u$ and $v$ along the normal direction at points of the surface $F(x,y,z)=0$; prove that

Find the envelope of the one-parameter family of plane curves: $x\cos \alpha +y\sin \alpha =p$ $(p=\text{constant})$.

Find the envelope of the one-parameter family of plane curves: $(x-a)^{2}+y^{2}=\dfrac {a^{2}}{2}$.

Find the envelope of the one-parameter family of plane curves: $y=kx+\dfrac {a}{k}$ $(a=\text{constant})$.

Find the envelope of the one-parameter family of plane curves: $y^{2}=2px+p^{2}$.

A segment of length $l$ has its two endpoints sliding along the coordinate axes; find the envelope of the family of segments so produced.

Find the envelope of the family of ellipses $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}=1$, given that the area $S$ of each ellipse in the family is constant.

A shell is fired in a vacuum with initial speed $v_{0}$; as the angle of elevation $\alpha$ varies in the vertical plane, find the envelope of the shell's trajectories.

Prove that the envelope of the normals of a plane curve is the evolute of that curve.

Investigate the properties of the discriminant curve of each of the following families of curves ($c$ being the parameter):

Find the envelope surface of the family of spheres of radius $r$ with centers on the circle $x=R\cos t$, $y=R\sin t$, $z=0$ ($t$ being the parameter, $R>r$).

Find the envelope surface of the family of spheres

(where $\cos^{2}\alpha +\cos^{2}\beta +\cos^{2}\gamma =1$, and $t$ is the parameter).

Find the envelope surface of the family of ellipsoids $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}+\dfrac {z^{2}}{c^{2}}=1$, where the volume $V$ of each ellipsoid is constant.

Find the envelope surface of the family of spheres of radius $\rho$ with centers on the cone $x^{2}+y^{2}=z^{2}$.

A luminous point is located at the origin of coordinates. If $x_{0}^{2}+y_{0}^{2}+z_{0}^{2}>R^{2}$, find the shadow cone produced by the projection of the sphere

If the parameters $p$ and $q$ are subject to the equation

find the envelope surface of the family of planes