Math Problems Vault

Problem 2335

Find the tangent plane to the surface $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}+\dfrac {z^{2}}{c^{2}}=1$ at the point $(x_{1},y_{1},z_{1})$ .

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Problem 2336

Find the tangent plane to the surface $x^{2}+y^{2}=z^{2}$ at the point $(x_{1},y_{1},z_{1})$ .

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Problem 2337

Find the tangent plane to the surface $xy^{2}+z^{3}=12$ at the point $(1,2,2)$ .

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Problem 2338

Find the tangent plane to the surface $x^{n}+y^{n}+z^{n}=a^{n}$ at the point $(x_{1},y_{1},z_{1})$ .

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Problem 2339

Find the tangent plane to the surface $(x^{2}+y^{2}+z^{2})^{2}=a^{2}(x^{2}-y^{2}+z^{2})$ at the point $(x_{1},y_{1},z_{1})$ .

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Problem 2340

The equation of an algebraic surface has the form $f(x,y,z)=0$ , where $f(x,y,z)$ is a polynomial. By introducing additionally powers of an auxiliary variable $t$ , this polynomial can be made homogeneous, after which the equation of the surface takes the form: $F(x,y,z,t)=0$ at $t=1$ . Prove that the equation of the tangent plane to this surface can be written in the form

\frac{\partial F}{\partial x}X+\frac{\partial F}{\partial y}Y+\frac{\partial F}{\partial z}Z+\frac{\partial F}{\partial t} = 0,

where, after differentiating, the variable $t$ is replaced by unity.

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Problem 2341

Determine the distance from the origin to the tangent plane of the surface $x\cos \dfrac {2\pi z}{h}=y\sin \dfrac {2\pi z}{h}$ .

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Problem 2342

Draw to the surface $xyz=1$ a tangent plane parallel to the plane $x+y+z-3=0$ .

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Problem 2343

(a)(b)

Find the lines along which the tangent plane to the surface $xy=az$ at the point $(x_{1},y_{1},z_{1})$ meets the surface:

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(a)

the line in the plane $x=x_{1}$ ;

(b)

the line in the plane $y=y_{1}$ .

Problem 2344

Find the equation of the tangent plane to the skew helicoid

x = u\cos v, \qquad y = u\sin v, \qquad z = av .

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Problem 2345

Find the equation of the tangent plane to the surface

x = u+v, \qquad y = u^{2}+v^{2}, \qquad z = u^{3}+v^{3}

at the point $(2,2,2)$ .

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Problem 2346

Prove that the tangent planes to the surface $xyz=a^{3}$ form, together with the coordinate planes, a tetrahedron of constant volume.

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Problem 2347

Prove that the planes tangent to the surface $\sqrt{x}+\sqrt{y}+\sqrt{z}=\sqrt{a}$ cut off, on the coordinate axes, segments whose sum is constant.

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Problem 2348

Prove that for the surface

x^{\frac{2}{3}}+y^{\frac{2}{3}}+z^{\frac{2}{3}} = a^{\frac{2}{3}}

the sum of the squares of the segments cut off on the axes by the tangent plane is a constant quantity.

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Problem 2349

Prove that the planes tangent to the surface $z=xf\left(\dfrac {y}{x}\right)$ all pass through one and the same point.

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Problem 2350

Prove that the planes tangent to the surface $z=x+f(y-z)$ are all parallel to one and the same line.

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Problem 2351

Prove that the segment on the axis $Oz$ cut off by the plane tangent to the surface $z+\sqrt{x^{2}+y^{2}+z^{2}}=x^{n}f\left(\dfrac {y}{x}\right)$ is proportional to the distance from the point of tangency to the origin.

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Problem 2352

Find the surface on which the tangents to a helix lie, and prove that the normal to this surface makes a constant angle with the axis $Oz$ .

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Problem 2353

Find the points on the ellipsoid $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}+\dfrac {z^{2}}{c^{2}}=1$ at which the normal makes a given constant angle $\alpha$ with the axis $Ox$ .

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Problem 2354

Prove that the point of intersection of the normal to the surface $x^{2}+y^{2}+z^{2}=xf\left(\dfrac {y}{x}\right)$ with the plane $xOy$ is equidistant from the origin and from the foot of the normal.

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Problem 2355

The segment of the normal to the surface $z^{2}=2a\sqrt{x^{2}+y^{2}}+b$ , lying between the surface and the plane $xOy$ , is projected onto $xOy$ . Prove that the length of the projection is constant.

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Problem 2356

Prove the same property (constancy of the length of the projection, onto the plane $xOy$ , of the segment of the normal lying between the surface and this plane) as in the preceding problem, for the normals to the surface

x = v\cos u-\varphi (u)\cos u+\varphi '(u)\sin u,

y = v\sin u-\varphi (u)\sin u-\varphi '(u)\cos u,

z = \sqrt{2v} .

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Problem 2357

From a point $M$ on the surface

x = r\cos \varphi , \qquad y = r\sin \varphi , \qquad z^{2} = -r^{2}+f(re^{\varphi })

a normal $MN$ is drawn to a point $N$ in the plane $xOy$ , and a perpendicular $MP$ is dropped onto the plane $xOy$ . Prove that the angle $NOP$ , where $O$ is the origin, equals $\dfrac {\pi }{4}$ .

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Problem 2358

Prove that for the surface $z^{2}=a^{2}\arctg \dfrac {y}{x}+f(x^{2}+y^{2})$ , the area of the triangle $NOP$ , obtained in the same way as in the preceding problem, is a constant quantity.

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Problem 2359

In the following problems one is required to establish the mutual orthogonality of the given surfaces (so that, if three surfaces are given, one must prove that each of them meets each of the others at a right angle). Prove the orthogonality of the following systems of surfaces:

xy = az^{2}, \qquad x^{2}+y^{2}+z^{2} = b, \qquad z^{2}+2x^{2} = c(z^{2}+2y^{2}) .

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Problem 2360

Prove the orthogonality of the following systems of surfaces (see problem 2359):

xyz = a^{3}, \qquad 2z^{2} = x^{2}+y^{2}+f(x^{2}-y^{2}) .

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Problem 2361

Prove the orthogonality of the following systems of surfaces (see problem 2359):

xy = \alpha z, \qquad \sqrt{x^{2}+z^{2}}+\sqrt{y^{2}+z^{2}} = \beta , \qquad \sqrt{x^{2}+z^{2}}-\sqrt{y^{2}+z^{2}} = \gamma .

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Problem 2362

Prove the orthogonality of the following systems of surfaces (see problem 2359):

x^{2}+y^{2}+z^{2} = ax, \qquad x^{2}+y^{2}+z^{2} = by, \qquad x^{2}+y^{2}+z^{2} = cz .

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Problem 2363

Prove the orthogonality of the following systems of surfaces (see problem 2359):

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

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Problem 2364

Through each point $(x,y,z)$ there pass three surfaces of the form $\dfrac {x^{2}}{a^{2}-\lambda }+\dfrac {y^{2}}{b^{2}-\lambda }+\dfrac {z^{2}}{c^{2}-\lambda }=1$ , for $\lambda =\lambda_{1}, \lambda_{2}, \lambda_{3}$ , representing respectively an ellipsoid, a hyperboloid of one sheet, and a hyperboloid of two sheets. Prove the orthogonality of these surfaces (see problem 2359).

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Problem 2365

From the points of the line

y = x, \qquad z = \frac{\pi }{4},

lying on the surface $y=x\tg z$ , normals are drawn to the surface. Prove that they lie on a hyperbolic paraboloid.

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Problem 2366

The same question (prove that the normals, drawn from the points of a given line lying on the surface, to the surface, lie on a hyperbolic paraboloid) as in the preceding problem, for the line

z = h, \qquad by = x\sqrt{a^{2}-h^{2}}

and the surface $b^{2}y^{2}=x^{2}(a^{2}-z^{2})$ .

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Problem 2367

Find the locus of the projections of the center of the ellipsoid $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}+\dfrac {z^{2}}{c^{2}}=1$ onto its tangent planes.

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Problem 2368

Along the ellipsoid of the preceding problem (with $a>b>c$ ), another, equal ellipsoid rolls without slipping or spinning, so that at one moment the endpoints of the major semi-axes were in contact. Find the locus of the center of the (rolling) ellipsoid.

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Problem 2369

Prove that the surface $x^{2}y^{2}+x^{2}z^{2}+y^{2}z^{2}=2xyz$ is met by the sphere $x^{2}+y^{2}+z^{2}=1$ in four circles.

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Problem 2370

Find the envelope of the following family of surfaces: spheres $(x-a)^{2}+y^{2}+z^{2}=1$ .

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Remark.

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Surfaces $f(x,y,z,a)=0$ , whose equations contain a coefficient $a$ varying from one surface to another, form a one-parameter family of surfaces. If two infinitely close surfaces meet in a certain curve, then the coordinates of the points of this curve, called the characteristic curve, satisfy the equations:

f(x,y,z,a) = 0, \qquad \frac{\partial f}{\partial a} = 0 .

Eliminating the parameter $a$ from these equations, we find an equation $F(x,y,z)=0$ . If the given family has an envelope surface, then it lies entirely on the surface obtained.

Similarly, the equation $f(x,y,z,a,b)=0$ represents a two-parameter family of surfaces. The limiting position of the point of intersection of three nearby surfaces

f(x,y,z,a,b) = 0, \qquad f(x,y,z,a+\Delta a,b) = 0, \qquad f(x,y,z,a,b+\Delta b) = 0

as $\Delta a\to 0$ and $\Delta b\to 0$ , i.e.the characteristic point of the family of surfaces, satisfies the equations:

f(x,y,z,a,b) = 0, \qquad \frac{\partial f}{\partial a} = 0, \qquad \frac{\partial f}{\partial b} = 0 .

Eliminating the parameters $a$ and $b$ from these, we obtain the equation of a surface $F(x,y,z)=0$ . If the given family has an envelope surface, then all its points satisfy the equation obtained; but other points may satisfy it as well.

Problem 2371

Find the envelope of the following family of surfaces (see problem 2370): planes passing through the point $(\sqrt{2},0,0)$ and at a distance equal to unity from the origin.

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Problem 2372

Find the envelope of the spheres

(x-lt)^{2}+(y-mt)^{2}+(z-nt)^{2} = a^{2},

where $t$ is a variable parameter.

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Problem 2373

Find the envelope of the spheres whose great circles lie on the paraboloid $z=x^{2}+y^{2}$ .

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Problem 2374

The same question (find the envelope of the spheres whose great circles lie on the paraboloid) as in the preceding problem, for the paraboloid $2z=\dfrac {x^{2}}{p}+\dfrac {y^{2}}{q}$ .

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Problem 2375

Find the envelope of the spheres of radius $a$ whose centers lie on the circle

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

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Problem 2376

Find the envelope of the planes tangent to the parabolas

y^{2} = 2x, \quad z = 0 \qquad \text{and} \qquad y^{2} = 2z, \quad x = 0 .

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Problem 2377

Find the envelope of the spheres

(x-lt)^{2}+(y-mt)^{2}+(z-nt)^{2} = p^{2}t^{2},

where $t$ is a variable parameter.

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Problem 2378

Find the envelope of the ellipsoids $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}+\dfrac {z^{2}}{c^{2}}=1$ subject to the condition $a^{2}+b^{2}+c^{2}=1$ .

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Problem 2379

The same (find the envelope of the ellipsoids $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}+\dfrac {z^{2}}{c^{2}}=1$ , see problem 2378) subject to the condition $a+b+c=1$ .

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Problem 2380

Find the envelope of the planes that, together with the coordinate planes, form a tetrahedron of constant volume.

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Problem 2381

Find the envelope of the planes for which the segments $a$ , $b$ , $c$ on the axes are related by the equation $a^{n}+b^{n}+c^{n}=1$ .

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Problem 2382

Show that the circles

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

are the characteristics of the family of spheres

x^{2}+y^{2}+z^{2}-2\bigl(f(t)+tf'(t)\bigr)x-2f'(t)y-a^{2} = 0 .

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