Circular Sections, Rectilinear Generators, and Other Problems

Through the line $2x = 2y = z$ draw a plane intersecting the surface $4x^{2}-y^{2}+z = 0$ in a rectangular hyperbola.

Find the vertex of the parabola obtained by cutting the cylinder $y^{2} = 2x$ with the plane $x+y+z = 1$.

Find the planes in which lie the axes of symmetry of the ellipses obtained by cutting the ellipsoid $x^{2}+2y^{2}+3z^{2} = 1$ with planes parallel to the plane $2x+y+z = 0$.

Find the equations of the axis of the parabola obtained in the section of the surface $x^{2}+y^{2}-2z^{2} = 1$ by the plane $x+y-2z = 1$.

Find the planes in which lie the axes of symmetry of the intersection of the surface $x^{2}+2y^{2}-2z = 0$ with planes parallel to the plane $x+y+z = 0$.

Find the plane in which lie the axes of symmetry of the parabolas obtained by intersecting the surface $y^{2}+2z^{2} = 2x$ with planes perpendicular to the vector $\overrightarrow {P}(0,1,-1)$.

Find the plane passing through the origin and through the axis of rotation of the surface

Find the equations of the planes intersecting the ellipsoid $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}+\dfrac {z^{2}}{c^{2}} = 1$ in circles. Here $a > b > c$.

Find the locus of the vertices of cones tangent along circles to the ellipsoid of the previous problem.

Find the circular sections of the hyperboloid of one sheet

for $a > b$.

Find the circular sections of the surface

Find the locus of the centers of the circular sections of the ellipsoid

Find the general form of the planes intersecting the surface

in circles.

Find the circular sections of the surface

For what $\lambda$ and $\mu$ are the circular sections of the surface $x^{2}+y^{2}-\lambda xz-\mu yz-\lambda az-a^{2} = 0$ perpendicular to the vector $\overrightarrow {P}(1,1,1)$?

Find the equation of the cylinder passing through the circle

and the point $(0,1,1)$ and having mutually perpendicular planes of circular section.

What is the locus of the centers of spheres of radius $R$ intersecting the ellipsoid $x^{2}+2y^{2}+3z^{2} = 1$ in circles?

Find the locus of the vertices of cones of revolution passing through the parabola

Prove that the locus of the vertices of circular cones passing through the ellipse

is the hyperbola

Hint. A sphere tangent to a circular cone and to some plane has its point of tangency with the plane at the focus of the resulting section.

Second-order surfaces with center at the point $(a,b,c)$ pass through the circle

Find the locus of the centers of the circular sections whose planes pass through the origin.

Find the equations of the lines along which the cone of revolution $xy+xz+yz = 0$ is intersected by the plane passing through the axis and the point $(1,2,3)$.

On the hyperbolic paraboloid $y^{2}-z^{2} = 2x$ find the locus of points through which two mutually perpendicular generators pass.

The same question for the surface $\dfrac {y^{2}}{p}-\dfrac {z^{2}}{q} = 2x$.

Find the rectilinear generators of the surface $xy+xz+x+y+1 = 0$.

The same question for the surface $x^{2}+3y^{2}+3z^{2}-2xy-2xz-2yz-6 = 0$.

Find the equations of the rectilinear generators of the surface $4y^{2}-z^{2}+x-8y-8z-2 = 0$ intersecting the axis $Ox$.

Find the magnitude of the largest angle between the generators of the cone $x^{2}+y^{2} = (x+y+z)^{2}$, as well as the direction of its axis.

Find the equation of the hyperboloid for which the lines

are generators of one family.

Find the equation of the paraboloid for which the lines

are generators of one family.

Find the general equation of the paraboloids passing through the circle

and having an axis parallel to the vector $\overrightarrow {P}(l,m,n)$.

Find the equation of the cone on whose surface lie the circles

Find the coordinates of the vertex of the cone on whose surface lie the circle

and the parabola

Find the paraboloid on which lie the points $(0,1,-1)$, $(1,-1,0)$ and the lines

Find the parabolic cylinder on which lie the parabolas

Find the paraboloid of revolution passing through the point $(1,1,2)$ and the circle

What is the locus of the centers of spheres of a given radius intersecting an elliptic paraboloid in circles?

Find the lengths of the axes of the ellipse obtained by cutting the ellipsoid $x^{2}+y^{2}+4z^{2} = 1$ with the plane $x+y+z = 0$.

Find the lengths of the axes of the ellipse obtained by cutting the paraboloid $2y^{2}+z^{2}-2x = 0$ with the plane $x = y$.

Find the parameter of the parabola lying on the surface $x^{2}+2y^{2}+z^{2}+4xy-2xz-4yz+2x-6z = 0$ and in the plane $x = z$.

A cone has its vertex at the focus of a prolate ellipsoid of revolution and rests on a plane section of the same ellipsoid. Prove that this cone is a cone of revolution.