Math Problems Vault

Problem 2228

Show that the curve

x = at^{2}+bt+c, \qquad y = a_{1}t^{2}+b_{1}t+c_{1}, \qquad z = a_{2}t^{2}+b_{2}t+c_{2}

lies in some plane, and find the equation of this plane.

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Problem 2229

For what relation among the coefficients $a$ , $b$ , $c$ , $a_{1}$ , $b_{1}$ , $c_{1}$ , $a_{2}$ , $b_{2}$ , $c_{2}$ does the curve

x = a\varphi (t)+b\psi (t)+c\omega (t)+d,

y = a_{1}\varphi (t)+b_{1}\psi (t)+c_{1}\omega (t)+d_{1},

z = a_{2}\varphi (t)+b_{2}\psi (t)+c_{2}\omega (t)+d_{2}

lie in a plane?

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Problem 2230

Show that the curve

x = \sin 2\varphi , \qquad y = 1-\cos 2\varphi , \qquad z = 2\cos \varphi

lies on the surface of a sphere.

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Problem 2231

Show the same (that the curve lies on the surface of a sphere) for the curve

x = \frac{t}{1+t^{2}+t^{4}}, \qquad y = \frac{t^{2}}{1+t^{2}+t^{4}}, \qquad z = \frac{t^{3}}{1+t^{2}+t^{4}} .

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Problem 2232

Convert to parametric form the equation of Viviani's curve, defined by the intersection of the sphere $x^{2}+y^{2}+z^{2}=a^{2}$ and the cylinder $x^{2}+y^{2}=ax$ , by setting $x=a\sin^{2}t$ .

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Problem 2233

Prove that the projection of the preceding curve onto the plane $xOz$ is a parabola.

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Problem 2234

Find the projection onto the plane $xOy$ of the curve of intersection of the paraboloid $z=x^{2}+y^{2}$ and the plane $x+y+z=1$ .

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Problem 2235

Find the projection of the curve

x = e^{t}\sin t, \qquad y = e^{t}\cos t, \qquad z = 2t

onto the plane $xOy$ .

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Problem 2236

Prove that the projection of the helix

x = a\cos t, \qquad y = a\sin t, \qquad z = bt

onto the plane $yOz$ is a sinusoid.

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Problem 2237

Prove that, upon translating the origin to the point $O_{1}(0,0,b\beta )$ and rotating the abscissa and ordinate axes about the axis $O_{1}z_{1}$ through an angle $\beta$ , the equations of the helix

x = a\cos t, \qquad y = a\sin t, \qquad z = bt

can be brought to the form

x_{1} = a\cos t_{1}, \qquad y_{1} = a\sin t_{1}, \qquad z_{1} = bt_{1} .

This shows that the helix is capable of sliding along itself.

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Problem 2238

Find the equation of the tangent to the curve

x = \frac{t^{4}}{4}, \qquad y = \frac{t^{3}}{3}, \qquad z = \frac{t^{2}}{2} .

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Problem 2239

Find the tangent to the preceding curve that is parallel to the plane $x+3y+2z=0$ .

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Problem 2240

Find the cosines of the angles with the axes for the tangent to the curve

x = t-\sin t, \qquad y = 1-\cos t, \qquad z = 4\sin \frac{t}{2} .

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Problem 2241

For the curve

x = -t\cos t+\sin t, \qquad y = t\sin t+\cos t, \qquad z = t+1

find the points at which the tangent is parallel to the plane $yOz$ or $xOz$ .

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Problem 2242

Find the tangent to the curve

x^{2}+y^{2} = 10, \qquad y^{2}+z^{2} = 25

at the point $(1,3,4)$ .

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Problem 2243

Find the tangent to the curve

2x^{2}+3y^{2}+z^{2} = 47, \qquad x^{2}+2y^{2} = z

at the point $(-2,1,6)$ .

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Problem 2244

Find the tangent to the curve

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

at the point $(m,m,2m^{2})$ .

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Problem 2245

Find the cosines of the angles with the axes for the tangent to the curve

x^{2} = 2az, \qquad y^{2} = 2bz .

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Problem 2246

The same question (find the cosines of the angles with the axes for the tangent) as in the preceding problem, for the circle

x^{2}+y^{2}+z^{2} = a^{2}, \qquad y = mx .

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Problem 2247

A tangent is drawn to the curve

y^{2} = 2px, \qquad z^{2} = 2qx

at the point where $x=\dfrac {p+q}{2}$ . Find its length from the point of tangency to the plane $x=0$ .

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Problem 2248

Find the angle between the tangent to the curve

x = a\cos t, \qquad y = a\sin t, \qquad z = bt

and the vector drawn from the origin to the point of tangency.

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Problem 2249

Find the normal plane to the curve

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

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Problem 2250

At the point where $x=\dfrac {p}{2}$ , find the normal plane to the curve

y^{2} = 2px, \qquad x^{2}+z^{2} = a^{2} .

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Problem 2251

Prove that the normal planes of the curve

x = a\cos t, \qquad y = a\sin \alpha \sin t, \qquad z = a\cos \alpha \sin t

pass through the line

x = 0, \qquad z+y\tg \alpha = 0 .

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Problem 2252

Prove that the curve

x = e^{t}\cos t, \qquad y = e^{t}\sin t, \qquad z = e^{t}

intersects all the generators of the cone $x^{2}+y^{2}=z^{2}$ at one and the same angle.

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Problem 2253

For what $a$ does the curve

x = e^{at}\cos t, \qquad y = e^{at}\sin t, \qquad z = e^{at}

intersect all the generators of the cone $x^{2}+y^{2}=z^{2}$ at an angle $\dfrac {\pi }{4}$ ?

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Problem 2254

Prove that the curves of intersection of the cylinders $y^{2}+z^{2}=m^{2}$ with the surface $xy=az$ intersect at a right angle all the generators of this surface belonging to one family.

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Problem 2255

The curve $x=a\tg t$ , $y=b\cos t$ , $z=b\sin t$ lies on the surface of a paraboloid. Prove that it intersects all the generators of one family at a right angle.

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Problem 2256

A curve called the loxodrome is defined by the equation $\varphi =a\ln \tg \left(\dfrac {\pi }{4}-\dfrac {\theta }{2}\right)$ , where $\theta$ is the latitude and $\varphi$ is the longitude of a point on a sphere. Prove that it intersects the meridians of the sphere at an angle whose tangent equals $a$ .

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Problem 2257

Prove that stereographic projection gives a conformal image, i.e.that lines in the plane intersect at the same angle as their images on the sphere.

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Remark.

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Stereographic projection is defined as follows. The topmost point of the sphere $x^{2}+y^{2}+z^{2}=az$ , i.e.the point $A(0,0,a)$ , is joined by a line to an arbitrary point $M(x,y,0)$ of the plane $xOy$ . The point $M_{1}$ , at which the line $AM$ meets the surface of the sphere $x^{2}+y^{2}+z^{2}=az$ , is taken as the image on the sphere of the point $M$ of the plane $xOy$ , and the point $M$ is the image of the point $M_{1}(x_{1},y_{1},z_{1})$ lying on the sphere. Thus stereographic projection gives a one-to-one correspondence between the points of the plane and of the sphere, with the exception of the point $A$ at the top of the sphere. Here

x_{1} = \frac{a^{2}x}{x^{2}+y^{2}+a^{2}}, \qquad y_{1} = \frac{a^{2}y}{x^{2}+y^{2}+a^{2}}, \qquad z_{1} = \frac{a(x^{2}+y^{2})}{x^{2}+y^{2}+a^{2}} .

Problem 2258

Prove that the curve $r=e^{m\varphi }$ , lying in the plane $xOy$ , where the polar axis coincides with the positive part of the axis $Ox$ , is taken by stereographic projection (see problem 2257) into a loxodrome.

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Problem 2259

Prove that circles on the sphere are taken by stereographic projection (see problem 2257) into circles in the plane, straight lines being regarded as a special case of circles.

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Problem 2260

The central projection of the sphere $x^{2}+y^{2}+z^{2}=2az$ onto the plane $xOy$ is formed as follows. An arbitrary point $M(x,y,0)$ of the plane $xOy$ is joined by a line $AM$ to the center of the sphere $A(0,0,a)$ . The point $M_{1}$ , at which $AM$ meets the sphere, is taken as the image of the point $M$ , and conversely. Prove that this projection is not a conformal image.

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Problem 2261

Prove that the tangent to the curve

x^{2} = 3y, \qquad 2xy = 9z

makes a constant angle with some fixed direction.

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Problem 2262

The coordinates of the points of a certain curve satisfy the relation

(x^{2}+y^{2}+z^{2}-a^{2})(dx^{2}+dy^{2}+dz^{2}) = (x\, dx+y\, dy+z\, dz)^{2} .

Prove that the tangents to this curve are tangent to the sphere

x^{2}+y^{2}+z^{2} = a^{2} .

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Problem 2263

Prove that the tangents to the curve

x = a(\sin t+\cos t), \qquad y = a(\sin t-\cos t), \qquad z = be^{-t}

meet the plane $xOy$ in the circle $x^{2}+y^{2}=4a^{2}$ .

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Problem 2264

(a)(b)

Find the curves along which the tangents to the helix

x = a\cos t, \qquad y = a\sin t, \qquad z = bt

meet:

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(a)

the plane $xOy$ ;

(b)

the plane $xOz$ .

Problem 2265

Find the spherical indicatrix of tangents of the helix

x = a\cos t, \qquad y = a\sin t, \qquad z = bt .

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Remark.

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The spherical indicatrix of tangents of a given curve is the locus of the endpoints of the unit vectors, drawn from the origin, that are parallel to these tangents.

Problem 2266

The same question (find the spherical indicatrix of tangents, see problem 2265) for the curve

x = 2t-\sin 2t, \qquad y = \cos 2t, \qquad z = 4\sin t .

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