Computing the tangent plane and normal at a point

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: $z=\operatorname {arctg}\dfrac {y}{x}$; 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: $z=y+\ln \dfrac {x}{z}$; 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 the tangent plane to the surface $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}+\dfrac {z^{2}}{c^{2}}=1$ at the point $(x_{1},y_{1},z_{1})$.

Find the tangent plane to the surface $x^{2}+y^{2}=z^{2}$ at the point $(x_{1},y_{1},z_{1})$.

Find the tangent plane to the surface $xy^{2}+z^{3}=12$ at the point $(1,2,2)$.

Find the tangent plane to the surface $x^{n}+y^{n}+z^{n}=a^{n}$ at the point $(x_{1},y_{1},z_{1})$.

Find the tangent plane to the surface $(x^{2}+y^{2}+z^{2})^{2}=a^{2}(x^{2}-y^{2}+z^{2})$ at the point $(x_{1},y_{1},z_{1})$.

The equation of an algebraic surface has the form $f(x,y,z)=0$, where $f(x,y,z)$ is a polynomial. By introducing additionally powers of an auxiliary variable $t$, this polynomial can be made homogeneous, after which the equation of the surface takes the form: $F(x,y,z,t)=0$ at $t=1$. Prove that the equation of the tangent plane to this surface can be written in the form

where, after differentiating, the variable $t$ is replaced by unity.

Determine the distance from the origin to the tangent plane of the surface $x\cos \dfrac {2\pi z}{h}=y\sin \dfrac {2\pi z}{h}$.

Draw to the surface $xyz=1$ a tangent plane parallel to the plane $x+y+z-3=0$.

Find the lines along which the tangent plane to the surface $xy=az$ at the point $(x_{1},y_{1},z_{1})$ meets the surface:

Find the equation of the tangent plane to the skew helicoid

Find the equation of the tangent plane to the surface

at the point $(2,2,2)$.

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

Prove that the planes tangent to the surface $\sqrt{x}+\sqrt{y}+\sqrt{z}=\sqrt{a}$ cut off, on the coordinate axes, segments whose sum is constant.

Prove that for the surface

the sum of the squares of the segments cut off on the axes by the tangent plane is a constant quantity.

Prove that the planes tangent to the surface $z=xf\left(\dfrac {y}{x}\right)$ all pass through one and the same point.

Prove that the planes tangent to the surface $z=x+f(y-z)$ are all parallel to one and the same line.

Prove that the segment on the axis $Oz$ cut off by the plane tangent to the surface $z+\sqrt{x^{2}+y^{2}+z^{2}}=x^{n}f\left(\dfrac {y}{x}\right)$ is proportional to the distance from the point of tangency to the origin.

Find the surface on which the tangents to a helix lie, and prove that the normal to this surface makes a constant angle with the axis $Oz$.

Find the points on the ellipsoid $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}+\dfrac {z^{2}}{c^{2}}=1$ at which the normal makes a given constant angle $\alpha$ with the axis $Ox$.

Prove that the point of intersection of the normal to the surface $x^{2}+y^{2}+z^{2}=xf\left(\dfrac {y}{x}\right)$ with the plane $xOy$ is equidistant from the origin and from the foot of the normal.

The segment of the normal to the surface $z^{2}=2a\sqrt{x^{2}+y^{2}}+b$, lying between the surface and the plane $xOy$, is projected onto $xOy$. Prove that the length of the projection is constant.

Prove the same property (constancy of the length of the projection, onto the plane $xOy$, of the segment of the normal lying between the surface and this plane) as in the preceding problem, for the normals to the surface

From a point $M$ on the surface

a normal $MN$ is drawn to a point $N$ in the plane $xOy$, and a perpendicular $MP$ is dropped onto the plane $xOy$. Prove that the angle $NOP$, where $O$ is the origin, equals $\dfrac {\pi }{4}$.

Prove that for the surface $z^{2}=a^{2}\arctg \dfrac {y}{x}+f(x^{2}+y^{2})$, the area of the triangle $NOP$, obtained in the same way as in the preceding problem, is a constant quantity.

From the points of the line

lying on the surface $y=x\tg z$, normals are drawn to the surface. Prove that they lie on a hyperbolic paraboloid.

The same question (prove that the normals, drawn from the points of a given line lying on the surface, to the surface, lie on a hyperbolic paraboloid) as in the preceding problem, for the line

and the surface $b^{2}y^{2}=x^{2}(a^{2}-z^{2})$.

Find the locus of the projections of the center of the ellipsoid $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}+\dfrac {z^{2}}{c^{2}}=1$ onto its tangent planes.

Along the ellipsoid of the preceding problem (with $a>b>c$), another, equal ellipsoid rolls without slipping or spinning, so that at one moment the endpoints of the major semi-axes were in contact. Find the locus of the center of the (rolling) ellipsoid.

Prove that the surface $x^{2}y^{2}+x^{2}z^{2}+y^{2}z^{2}=2xyz$ is met by the sphere $x^{2}+y^{2}+z^{2}=1$ in four circles.

Write the equations of the tangent plane and the normal to the surface at the given point.

Write the equations of the tangent plane and the normal to the surface at the given point.

Write the equations of the tangent plane and the normal to the surface at the given point.

Write the equation of the tangent plane to the surface at its given point $\left(x_{0} ; y_{0} ; z_{0}\right)$:

Write the equation of the tangent plane to the surface:

Write the equation of the tangent plane to the surface:

Find the distance from the origin to the tangent plane to the surface $z = \operatorname {arctg}(y / x)$ at its point $\left(x_{0} ; y_{0} ; z_{0}\right)$.

Find the angles that the normal to the surface

at the point $(1 / 4 ; 1 / 4 ; \pi / 4)$ makes with the coordinate axes.