Math Problems Vault

Problem 10.46

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(1)

The sections of the surface $x^{2} + 2y^{2} - 3z^{2} - 1 = 0$ by the planes $x = 0, x = 1, x = 2$ are projected onto the plane $Oyz$ . Depict the projections.

(2)

The sections of the surface $x^{2} + 2y^{2} - 3z^{2} = 0$ by the planes $x = - 1, x = 0, x = 1$ are projected onto the plane $Oyz$ . Depict the projections.

(3)

The sections of the surface $2x^{2} - y^{2} = 2z$ by the planes $x = - 1$ , $x = 0, x = 1$ are projected onto the plane $Oyz$ . Depict the projections.

(4)

The sections of the surface $2x^{2} - y^{2} = 2z$ by the planes $y = - 1$ , $y = 0, y = 1$ are projected onto the plane $Oxz$ . Depict the projections.

(5)

The sections of the surface $2x^{2} - y^{2} = 2z$ by the planes $z = - 1$ , $z = 0, z = 1$ are projected onto the plane $Oxy$ . Depict the projections.

Problem 10.47

(1)(2)

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(1)

The sections of the surfaces $x^{2} + 2y^{2} - 3z^{2} - 1 = 0, x^{2} + 2y^{2} - 3z^{2} = 0, x^{2} + 2y^{2} - 3z^{2} + 1 = 0$ by the plane $x = 0$ are projected onto the plane $Oyz$ . Depict the projections.

(2)

The sections of the same surfaces by the plane $z = 1$ are projected onto the plane $Oxy$ . Depict the projections.

Problem 10.48

(1)(2)

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(1)

Is the curve of intersection of two surfaces of the second order a plane curve? Give examples.

(2)

Suppose the curve of intersection of two surfaces of the second order is plane. Will this curve be algebraic? If so, of what order? Consider examples.

Problem 10.49

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.

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Problem 10.50

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.

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Problem 10.51

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:

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(1)

$\mathcal{F}$ is a cylindrical surface with generator parallel to the axis $Oz$ and directrix $\mathcal{L}$

(2)

$\mathcal{L}$ is the section of $\mathcal{F}$ by the plane $Oxy$

(3)

$\mathcal{L}$ is the projection of $\mathcal{F}$ onto the plane $Oxy$

(4)

$\mathcal{L}$ is the projection of any directrix of the cylinder onto the plane $Oxy$

(5)

$\mathcal{L}$ contains the projection onto the plane $Oxy$ of any curve lying on the cylinder $\mathcal{F}$ .

Problem 10.52

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?

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Problem 10.53

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.

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Problem 10.54

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.

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Problem 10.55

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

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Problem 10.56

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

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Problem 10.57

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

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Problem 10.58

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

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Problem 10.59

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.

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Problem 10.60

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.

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Problem 10.61

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.

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Problem 10.62

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

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Problem 10.63

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 $x^{2} + y^{2} + z^{2} = 9, x - z = 0$ .

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Problem 10.64

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

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Problem 10.65

(1)(2)

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

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(1)

$x + y + 2z = 5$

(2)

$x + y + z = 7$ .

Problem 10.66

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

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Problem 10.67

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.

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Problem 10.68

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 $x + y + z = 1$ .

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Problem 10.69

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 $x + y + z = 1$ .

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Problem 10.70

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 $x + y + z = 1$ .

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Problem 10.71

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 $C(3, 2, 1)$ .

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Problem 10.72

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?

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Problem 10.73

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?

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Problem 10.74

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.

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Problem 10.75

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.

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Problem 10.76

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.

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