Equations of conical and cylindrical surfaces

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Problem 2311 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 5.9 "Surfaces. Their Equations"

Surfaces traced out by a line passing through a given point (a,b,c)(a,b,c) and a given curve are called conical. Prove that their equations can be represented in the form

f(x−az−c,y−bz−c)=0. f\left(\frac{x-a}{z-c}, \frac{y-b}{z-c}\right) = 0 .
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Problem 2312 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 5.9 "Surfaces. Their Equations"

Surfaces formed by the motion of a line parallel to a vector P→(l,m,n)\overrightarrow {P}(l,m,n) and passing through a given curve are called cylindrical. Prove that the general form of the equation of cylindrical surfaces is

f(nx−lz,ny−mz)=0. f(nx-lz, ny-mz) = 0 .
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Problem 2315 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 5.9 "Surfaces. Their Equations"

Find the equation of the cylinder circumscribed about the sphere x2+y2+z2=a2x^{2}+y^{2}+z^{2}=a^{2}, with generators parallel to the line

x=lt,y=mt,z=nt. x = lt, \qquad y = mt, \qquad z = nt .
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Problem 2316 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 5.9 "Surfaces. Their Equations"

Find the equation of the cylinder with generators parallel to the line x=y=zx=y=z, circumscribed about the ellipsoid x2+4y2+9z2=1x^{2}+4y^{2}+9z^{2}=1.

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Problem 2317 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 5.9 "Surfaces. Their Equations"

Find the equation of the cylindrical surface whose directrix is the curve

z=0,x2+y2=ay, z = 0, \qquad x^{2}+y^{2} = ay,

with generators parallel to the vector P→(l,m,n)\overrightarrow {P}(l,m,n).

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Problem 2318 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 5.9 "Surfaces. Their Equations"

Find the equation of the conical surface with vertex at the point (−1,0,0)(-1,0,0) and circumscribed about the paraboloid 2y2+z2=4x2y^{2}+z^{2}=4x.

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Problem 2319 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 5.9 "Surfaces. Their Equations"

Find the equation of the conical surface with vertex at the point (a,b,c)(a,b,c), whose directrix is the parabola

z=0,y2=2px. z = 0, \qquad y^{2} = 2px .
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Problem 2320 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 5.9 "Surfaces. Their Equations"

Find the equation of the conical surface with vertex at the point (0,0,−c)(0,0,-c), whose directrix is the lemniscate

z=0,(x2+y2)2=a2(x2−y). z = 0, \qquad (x^{2}+y^{2})^{2} = a^{2}(x^{2}-y) .
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Problem 2321 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 5.9 "Surfaces. Their Equations"

Circumscribe about the paraboloid x2+y2=2zx^{2}+y^{2}=2z a conical surface with vertex at the point (0,0,−2)(0,0,-2).

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Problem 2322 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 5.9 "Surfaces. Their Equations"

Find the equation of the conical surface circumscribed about the surface xyz=c3xyz=c^{3} with vertex at (0,0,−3c)(0,0,-3c).

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References

[ГюКу03] Гюнтер Н.М., Кузьмин Р.О. Сборник задач по высшей математике. 13-е изд., Издательство «Лань», 2003.