Set equations in space and elementary theory of second-order surfaces

Can an algebraic surface of the second order be a line? A plane? The empty set? Give examples.

A family of surfaces is given in a rectangular coordinate system by an equation containing an arbitrary parameter $\lambda$. Determine the type of the surface for all possible values of $\lambda$:

Write the equation of the sphere:

Find the coordinates of the center $C$ and the radius $R$ of the sphere:

Find the coordinates of the center of the surface, its semi-axes, and the equations of the planes of symmetry, and depict the surface in the original coordinate system:

Find the coordinates of the center of the surface, its vertices, the equations of the axis of symmetry and the planes of symmetry, and depict the surface in the original coordinate system:

Determine the type of the surface:

Determine the type of the surface, depict the surface in the original coordinate system:

Find the axis of revolution of the surface, depict the surface $x^{2} + z^{2} + x = 0$.

Determine the type of the surface, find the axis of revolution, the coordinates of the vertices, and depict the surface:

Find the coordinates of the center of the surface, the equations of the axis of revolution and of the throat circle, determine the radius of the throat circle, and depict the surface $x^{2} + 2yz = 1$.

Find the points of intersection of the surface $x^{2} + y^{2} = z$ and the line:

How many common points can a line and a surface of the second order have? Give examples.

Determine whether the point $M(1, 1, 1)$ lies inside or outside the ellipsoid $x^{2} + 2y^{2} + 3z^{2} = 4$.

The axis $Oz$ points upward. Determine whether the point $M(1, 1, 1)$ lies above or below the paraboloid $x^{2} + 2y^{2} = 2z$.

Give examples of surfaces of revolution that are algebraic surfaces of order $2, 3, 4$.

Name the types and write out the canonical equations of cylindrical surfaces of the second order.

Give examples of cylindrical surfaces that are algebraic surfaces of order 3, 4.

Give an example of a cylindrical surface that is not algebraic.

Give examples of cylinders and cones that are not surfaces of revolution.

Prove that every equation of the form $F(x, y, z) = 0$, where $F$ is a homogeneous polynomial, defines a cone with vertex at the origin.

Give an example of a conical surface that is not algebraic.

Can a plane be regarded as a special case of a conical surface? As a special case of a cylinder? As a surface of revolution?

Find the vector equation of the right circular cylinder of radius $R$ having axis $\mathbf{r} = \mathbf{r}_{0} + \mathbf{a} t$.

Find the vector equation of the sphere with center at the point $M_{0}\left(\mathbf{r}_{0}\right)$ and radius $R$.

Find the vector equation of the right circular cone with vertex at the point $M_{0}\left(\mathbf{r}_{0}\right)$ and axis $\mathbf{r} = \mathbf{r}_{0} + \mathbf{a} t$, given that the angle between its generator and the axis equals $\alpha$.

Find the vector equation of the ellipsoid obtained by rotating an ellipse with foci at the points $M_{1}\left(\mathbf{r}_{1}\right), M_{2}\left(\mathbf{r}_{2}\right)$ and major semi-axis $a$ about the major axis of the ellipse.

Find the equation of the surface obtained by rotating the parabola $z^{2} = x$:

Find the equation and determine the type of the surface obtained by rotating the hyperbola $x^{2} - y^{2} = 2$:

Find the equation of the surface obtained by rotating the circle $x^{2} + y^{2} - 4x + 3 = 0$ about the axis $Oy$.

Find the equations of the surfaces obtained by rotating the hyperbola $xy = 1$ about the asymptotes.

Prove that the cylindrical surface with directrix given by the parametric equations $x = \varphi (t), y = \psi (t), z = \chi (t)$, and with generator parallel to the vector $\mathbf{a}\left(a_{1}, a_{2}, a_{3}\right)$, is defined by the equations $x = \varphi (u) + a_{1} v$, $y = \psi (u) + a_{2} v, z = \chi (u) + a_{3} v$.

Prove that the cone with directrix given by the parametric equations $x = \varphi (t), y = \psi (t), z = \chi (t)$, and with vertex at the origin, is defined by the equations $x = u\varphi (v), y = u\psi (v), z = u\chi (v)$.

Find the equation of the right circular cylinder of radius $\sqrt{2}$ with axis $x = 1 + t, y = 2 + t, z = 3 + t$.

Find the equation of the right circular cylinder passing through the point $M(1, 1, 2)$ and having axis $x = 1 + t$, $y = 2 + t, z = 3 + t$.

Find the equation of the right circular cone with vertex at the origin and direction of axis determined by the vector $\mathbf{a}(1, 1, 1)$, given that the generators of the cone make an angle $\arccos (1 / \sqrt{3})$ with its axis.

Find the equation and determine the type of the surface formed by rotating the line $x = 1 + t, y = z = 3 + t$ about the axis $\mathrm{Oz}$.

Find the equation and determine the type of the surface formed by rotating the line $x = 0, y - z + 1 = 0$ about the axis $\mathrm{Oz}$.

Find the equation and determine the type of the surface formed by rotating the line $x = - t, y = z = 2t$ about the line $x = y = z$.

Find the equation of the cone with vertex at the point $M(1, 1, 1)$, tangent to the sphere $x^{2} + y^{2} + z^{2} = 2$.

Find the parametric equations of the cylinder with generator parallel to the vector $\mathbf{a}(1, 1, 1)$, and directrix given by the equations $x = - 1 + 2\cos t, y = - 1 + 2\sin t, z = 3 - 2\cos t - 2\sin t$.

Eliminating the parameters, obtain the algebraic equation of the surface $x = u + \cos v, \quad y = u + \sin v, \quad z = u - \cos v - \sin v$. What surface is this?

Determine the shape of the curve of intersection of the surfaces $x^{2} + y^{2} = 2z, x^{2} + y^{2} + z^{2} = 8$ and find its parametric equations.

Prove that the curve of intersection of a surface of the second order with a plane is an algebraic curve of order at most two. Give examples where this is a curve of the first order.

Let $\mathcal{F}$ be the surface defined by the algebraic equation $F(x, y) = 0$, and let $\mathcal{L}$ be the nonempty set of points defined by the equations $F(x, y) = 0, z = 0$. Prove the statements:

Find the equation of the projection of the curve of intersection of the surfaces $x^{2} + 2y^{2} = 2z, x + 2y + z = 1$ onto the plane $Oxy$. What is this curve?

Let $\mathcal{S}$ be the section of the paraboloid $x^{2} + y^{2} = 2z$ by a plane that meets the positive semi-axis $\mathrm{Oz}$ at a single point. Prove that the projection of $\mathcal{S}$ onto the plane $Oxy$ is a circle.

Prove that the curve of intersection of the surfaces $x^{2} + y^{2} = 2z, x + y + z = 1$ is an ellipse, and find its parametric equations.

Along what curve do the paraboloid $x^{2} - y^{2} = 2z$ and the plane $x + y + z = 1$ intersect?

Find the coordinates of the center and the radius of the circle $x^{2} + y^{2} + z^{2} - 12x + 4y - 6z + 24 = 0, 2x + 2y + z + 1 = 0$.

Find the parametric equations of the cone with vertex at the origin and directrix defined by the equations $x^{2} + y^{2} = 2z, x + y + z = 1$.

Find the equation of the cylinder with generator parallel to the axis $Oz$, and directrix the circle $x^{2} + y^{2} + z^{2} = 3, z = 1$.

The generator of a cylinder is parallel to the axis $Oz$, its directrix is the circle $x^{2} + y^{2} = 2z, x^{2} + y^{2} + z^{2} = 8$. Find the equation of the cylinder.

The generators of a cylinder are parallel to the axis $\mathrm{O} z$, its directrix is the ellipse $x^{2} + y^{2} = 2z, x + y + z = 1$. Prove that this is a right circular cylinder, write its equation, and find its axis and radius.

The generators of a cylinder are parallel to the vector $\mathbf{a}(1, 1, 1)$, its directrix is the circle $x^{2} + y^{2} = 2z$, $x^{2} + y^{2} + z^{2} = 8$. Write the equation of the cylinder.

Find the equation of the cone with vertex at the point $O(0, 0, 0)$ and directrix the circle $x^{2} + y^{2} + z^{2} = 1, x + y + z = 1$.

Find the equation of the ellipsoid whose planes of symmetry coincide with the coordinate planes, containing the point $M(3, 1, 1)$ and the circle $x^{2} + y^{2} + z^{2} = 9, x - z = 0$.

Prove that the centers of the plane sections of an elliptic cylinder lie on its axis.

Find the center of the section of the ellipsoid $x^{2} + 2y^{2} + 4z^{2} = 40$ by the plane:

Find the center of the section of the hyperboloid $x^{2} + 2y^{2} - 4z^{2} = - 4$ by the plane $x + y + 2z = 2$.

Let $M_{0}(5, 7, 20)$ be a point of a plane, and let $\mathbf{p}(- 3 / \sqrt{11}$, $1 / \sqrt{11}, 1 / \sqrt{11}), \quad q(1 / \sqrt{6}, 1 / \sqrt{6}, 2 / \sqrt{6})$ be an orthonormal basis in it. Write the equations of the curve of intersection of this plane and the cone $x^{2} + 5y^{2} - z^{2} = 0$ in the internal coordinate system $M_{0}, \mathbf{p}, \mathbf{q}$. Find the coordinates of the center of the curve of intersection and the equations of its axes of symmetry in the original (spatial) coordinate system.

Find the equation of the set of centers of the sections of the ellipsoid $x^{2} + 2y^{2} + 3z^{2} = 4$ by planes parallel to the plane $x + y + z = 1$.

Find the equation of the set of centers of the sections of the hyperboloid $x^{2} + y^{2} - 3z^{2} = 2$ by planes parallel to the plane $x + y + z = 1$.

Find the equation of the set of centers of the sections of the paraboloid $x^{2} + y^{2} = 2z$ by planes parallel to the plane $x + y + z = 1$.

Find the equation of the plane that intersects the ellipsoid $x^{2} + 2y^{2} + 4z^{2} = 9$ in an ellipse whose center is at the point $C(3, 2, 1)$.

Find the equations of the projections of the curve of intersection of the ellipsoid $3x^{2} + 4y^{2} + 5z^{2} = 36$ and the sphere $x^{2} + y^{2} + z^{2} = 9$ onto the coordinate planes. What is this section?

Find the equations of the projections of the curve of intersection of the ellipsoids $x^{2} + 2y^{2} + 3z^{2} = 4, 3x^{2} + 5y^{2} + 6z^{2} = 10$ onto the coordinate planes. What is this curve?

Write the equations of the projections of the curve of intersection of the surfaces $x^{2} + y^{2} - z^{2} = 1, x^{2} - y^{2} = 2z$ onto the coordinate planes. What is this curve? Find its parametric equations.

Find the equations of the projections of the curve of intersection of the surfaces $5x^{2} - 3y^{2} + 4z^{2} = 0, x^{2} - y^{2} + z^{2} + 1 = 0$ onto the coordinate planes.

Prove that the curve of intersection of the paraboloid $x^{2} + 2y^{2} = 4z + 10$ and the sphere $x^{2} + y^{2} + z^{2} = 6$ consists of two circles. Find the points of intersection of these circles and their radii.

Name the types of surfaces of the second order that have rectilinear generators.

Can the number of rectilinear generators passing through a single point of a surface of the second order equal 0? 1? 2? 3?.. Can it be infinite? Give examples.

Find the equation of the family of rectilinear generators of the cylinder $x^{2} - y^{2} = 1$.

Find the equation of the family of rectilinear generators of the cone $x^{2} + y^{2} - z^{2} = 0$.

Find the rectilinear generators of the paraboloid $4x^{2} - y^{2} = 16z$ that meet at the point $M(2, 0, 1)$.

Find the equation of the plane passing through the points $M(1, 1, 1)$ and $N(2, 0, 2)$ and intersecting the paraboloid $x^{2} - y^{2} = 2z$ in a pair of lines.

Find the equation of the plane that intersects the hyperboloid $x^{2} + 4y^{2} - 9z^{2} = 36$ in a pair of lines passing through the point $M(6, - 3, 2)$.

Given the paraboloid $x^{2} - y^{2} = 2z$ and the plane $x + y + z = 1$. Find the equation of the plane parallel to the given one that intersects the paraboloid in a pair of lines. Find the equations of these lines and the angle between them.

Two rectilinear generators of the hyperboloid of revolution $x^{2} + y^{2} - z^{2} = 1$ meet at a point belonging to the plane $z = h$. Find the angle between them:

Find the set of points of the surface $\mathcal{S}$ at which its mutually orthogonal rectilinear generators intersect, if $\mathcal{S}$ is defined by the equation:

Prove that the projections of the rectilinear generators of the paraboloid $x^{2} - y^{2} = 2z$ onto the plane $Oxz$ are tangent to the parabola $x^{2} = 2z$.

Prove that the projections of the rectilinear generators of the hyperboloid $x^{2} + y^{2} - z^{2} = 1$ onto the plane $Oxy$ are tangent to the circle $x^{2} + y^{2} = 1$.