Math Problems Vault

Problem 2311

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

f\left(\frac{x-a}{z-c}, \frac{y-b}{z-c}\right) = 0 .

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Problem 2312

Surfaces formed by the motion of a line parallel to a vector $\overrightarrow {P}(l,m,n)$ 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 .

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Problem 2313

Prove that the equation of a surface of revolution about the axis $Oz$ has the form $f(x^{2}+y^{2}, z)=0$ .

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Problem 2314

Prove that the equation of a surface of revolution about the line

x = a+lt, \qquad y = b+mt, \qquad z = c+nt

has the form

f(lx+my+nz, R) = 0,

where

R = \bigl(n(y-b)-m(z-c)\bigr)^{2}+\bigl(l(z-c)-n(x-a)\bigr)^{2}+{}

{}+\bigl(m(x-a)-l(y-b)\bigr)^{2} .

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Problem 2315

Find the equation of the cylinder circumscribed about the sphere $x^{2}+y^{2}+z^{2}=a^{2}$ , with generators parallel to the line

x = lt, \qquad y = mt, \qquad z = nt .

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Problem 2316

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

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Problem 2317

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

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

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

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Problem 2318

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

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Problem 2319

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

z = 0, \qquad y^{2} = 2px .

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Problem 2320

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

z = 0, \qquad (x^{2}+y^{2})^{2} = a^{2}(x^{2}-y) .

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Problem 2321

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

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Problem 2322

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

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Problem 2323

Find the equation of the surface obtained by rotating the ellipse $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}=1$ about the axis $Ox$ .

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Problem 2324

Find the equation of the surface obtained by rotating the hyperbola $\dfrac {x^{2}}{a^{2}}-\dfrac {y^{2}}{b^{2}}=1$ about the axis $Oy$ .

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Problem 2325

Find the equation of the surface obtained by rotating the parabola $y^{2}=2px$ about the axis $Ox$ .

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Problem 2326

Find the equation of the surface obtained by rotating the circle $(x-a)^{2}+y^{2}=R^{2}$ about the axis $Oy$ .

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Problem 2327

Find the equation of the surface obtained by rotating the parabola $y^{2}=2px$ about the axis $Oy$ .

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Problem 2328

Find the equation of the surface obtained by rotating the lemniscate $(x^{2}+y^{2})^{2}=a^{2}(x^{2}-y^{2})$ about $Ox$ .

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Problem 2329

Find the equation of the surface obtained by rotating the same curve (the lemniscate $(x^{2}+y^{2})^{2}=a^{2}(x^{2}-y^{2})$ , see problem 2328) about the axis $Oy$ .

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Problem 2330

The surface obtained by rotating the sinusoid

x = \sin z, \qquad y = 0

about the axis $Oz$ is illuminated by parallel rays making an angle $\dfrac {\pi }{4}$ with $Oz$ . Find the shape of the shadow it casts on the plane $xOy$ .

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Problem 2331

The parametric representation of surfaces, introduced by Gauss, consists in giving the coordinates of the points of the surface as functions of two parameters: $x=\varphi (u,v)$ , $y=\psi (u,v)$ , $z=\omega (u,v)$ .

The equation of a sphere can be represented in the form

x = a\sin \theta \cos \varphi , \qquad y = a\sin \theta \sin \varphi , \qquad z = a\cos \theta .

Find a similar form of the equation for an ellipsoid.

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Problem 2332

A surface is given by the equations

x = a\cos ^{4}u\cos ^{4}v, \qquad y = a\cos ^{4}u\sin ^{4}v, \qquad z = a\sin ^{4}u .

Find its equation in ordinary form.

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Problem 2333

The same question (find the equation in ordinary form) for the surface given by the equations

x = a\cos ^{3}u\cos ^{3}v, \qquad y = a\cos ^{3}u\sin ^{3}v, \qquad z = a\sin ^{3}v .

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Problem 2334

The same question (find the equation in ordinary form) for the surface given by the equations

x = a\frac{u^{2}+v^{2}-1}{u^{2}+v^{2}+1}, \qquad y = b\frac{2u}{u^{2}+v^{2}+1}, \qquad z = c\frac{2v}{u^{2}+v^{2}+1} .

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