Tangent Planes and Lines to Second-Order Surfaces

Prove that the plane tangent to the ellipsoid $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}+\dfrac {z^{2}}{c^{2}} = 1$ at the point $(x_{1},y_{1},z_{1})$ can be represented by the equation

Prove that the plane tangent to the hyperboloid $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}-\dfrac {z^{2}}{c^{2}} = \sigma$ at the point $(x_{1},y_{1},z_{1})$ has the equation

Prove that the plane tangent at the point $(x_{1},y_{1},z_{1})$ to the paraboloid $\dfrac {x^{2}}{a^{2}} \pm \dfrac {y^{2}}{b^{2}} = 2z$ has the equation

Find the equation of the plane tangent to the ellipsoid

and cutting off on the axes segments whose ratio is $a:b:c$.

The same question for the hyperboloid $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}-\dfrac {z^{2}}{c^{2}} = 1$.

On the ellipsoid $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}+\dfrac {z^{2}}{c^{2}} = 1$ find a point such that the tangent plane there cuts off segments of equal length on the coordinate axes.

On the tangent plane to the ellipsoid of the previous problem, the coordinate planes form a triangle with centroid at the point of tangency. Find the coordinates of the point of tangency.

Prove that when $a^{2}+b^{2}+c^{2} = l^{2}$ the ellipsoid

is inscribed in the octahedron $\left|x\right|+\left|y\right|+\left|z\right| = l$.

Prove that all the normals to the surface $xy+xz+yz = 0$ make an equal angle with the line $x = y = z$.

Prove the same for the surface $x^{2}+y^{2}+z^{2} = xy+xz+yz+1$.

On the surface $xy+xz+yz = 3$ find the point closest to the plane $x+y+z = 0$.

On the surface $x^{2}+y^{2}+z^{2}+xy+xz+yz = 6$ find the highest and lowest points.

On the hyperbolic paraboloid $xy = z$ find the points at which the normal makes an angle of $45^{\circ }$ with $Oz$.

On the paraboloid $x^{2}-y^{2} = az$ a curve is drawn such that at every point of it the normal to the surface makes a constant angle with $Oz$. Prove that the projection of this curve onto $xOy$ is a circle centered at the origin.

Prove that the plane tangent to the surface $x^{2}+y^{2}+z^{2} = xy+xz+yz$ at the point $M(x_{1},y_{1},z_{1})$ contains the line drawn from the origin to the point $M$.

The plane tangent to the hyperboloid $x^{2}+y^{2}-z^{2} = 1$ at the point $(1,0,0)$ intersects it in two lines. Determine the angle between them.

The same question for the paraboloid $x^{2}-y^{2} = z$ and the point $(0,0,0)$.

Through the line

draw a plane tangent to the ellipsoid $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}+\dfrac {z^{2}}{c^{2}} = 1$.

Cones are tangent to the ellipsoid of the previous problem along plane curves whose planes are parallel to the plane $x+y+z = 0$. Prove that the locus of the vertices of the cones is the line

Prove that the condition of tangency of the plane $z = \alpha x+\beta y+\gamma$ and the surface $Ax^{2}+By^{2}-2Cz = 0$ is the equality

Prove that the plane parallel to the plane $z = \alpha x+\beta y$ and tangent to the ellipsoid $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}+\dfrac {z^{2}}{c^{2}} = 1$ has the equation

Prove that the condition of tangency of the line

and the surface $Ax^{2}+By^{2}+Cz^{2}+D = 0$ is expressed by the equality

If, in addition, the equalities

hold, then the line lies on the surface.

A line parallel to the vector $\overrightarrow {P}(l,m,n)$ is tangent at the point $(x_{1},y_{1},z_{1})$ to the surface $Ax^{2}+By^{2}+Cz^{2}+D = 0$. Prove that $Alx_{1}+Bmy_{1}+Cnz_{1} = 0$.

A plane perpendicular to the vector $\overrightarrow {P}(l,m,n)$ is tangent to the surface $Ax^{2}+By^{2}+Cz^{2}+D = 0$. Show that its equation has the form

A plane perpendicular to the vector $\overrightarrow {P}(l,m,n)$ is tangent to the surface $Ax^{2}+By^{2}+z = 0$. Show that its equation has the form

Prove that the locus of the vertices of right trihedral angles tangent by their faces to the surface $Ax^{2}+By^{2}+Cz^{2}+D = 0$ is the spherical surface

assuming the right-hand side is positive.

Prove that the locus of the vertices of right trihedral angles tangent by their faces to the surface $Ax^{2}+By^{2}+z = 0$ is the plane $z = \dfrac {1}{4A}+\dfrac {1}{4B}$.

An ellipsoid with semi-axes $a$, $b$, and $c$ slides while remaining tangent to the three coordinate planes. Prove that the distance of its center from the origin equals $\sqrt{a^{2}+b^{2}+c^{2}}$.

Prove that the locus of the vertices of right trihedral angles tangent by their edges to the surface $Ax^{2}+By^{2}+Cz^{2}+D = 0$ is the surface

A right trihedral angle slides while remaining tangent by its edges to the surface $Ax^{2}+By^{2}+2z = 0$. Prove that its vertex moves on the surface

Find the minimal dimensions of a box with walls parallel to the coordinate planes, into which the ellipsoid $(x+y+z)^{2}+x^{2}+y^{2}+z^{2} = a^{2}$ can be inscribed.

The sphere $x^{2}+y^{2}+z^{2} = 2z$ is illuminated by a beam of rays parallel to the line $x = y = z$. Find the shape of the shadow on the plane $xOy$.

Find the shadow of the ellipsoid $(x+y+z)^{2}+x^{2}+y^{2}+z^{2} = a^{2}$ on the plane $x+y+z = 0$, if the rays are perpendicular to this plane.

The paraboloid $xy = z$ is projected onto the plane $xOy$ by rays parallel to the line $x = y = z$. Find the shape of the shadow.

The points of tangency of planes with a second-order surface lie on the line of intersection of the surface with a plane passing through the center. Prove that the tangent planes are parallel to one and the same line.

Around the ellipsoid $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}+\dfrac {z^{2}}{c^{2}} = 1$, where $a \geqslant b \geqslant c$, $a > c$, it is possible to circumscribe a cylinder with circular cross-section. Find the equation of the cylinder.

Prove that as points move along some plane, the polar plane of a point with respect to a given second-order surface rotates about the corresponding point.

Prove that as a point moves along a line, its polar plane rotates about the corresponding line.

Find the locus of the points of tangency of the tangent lines drawn from a given point to a given second-order surface.

Find the locus of the points of tangency of the tangents to a given second-order surface that are parallel to the vector $\overrightarrow {P}(l,m,n)$.

A circle of radius $a$ slides while remaining tangent to the three coordinate planes. Find the surface traced out by its center.

A right trihedral angle is tangent by its faces to a given circle of radius $a$. Prove that its vertex traces out a spherical surface of radius $a\sqrt{2}$.