Envelopes of families of curves and surfaces

Find the envelope of the one-parameter family of plane curves: $x\cos \alpha +y\sin \alpha =p$ $(p=\text{constant})$.

Find the envelope of the one-parameter family of plane curves: $(x-a)^{2}+y^{2}=\dfrac {a^{2}}{2}$.

Find the envelope of the one-parameter family of plane curves: $y=kx+\dfrac {a}{k}$ $(a=\text{constant})$.

Find the envelope of the one-parameter family of plane curves: $y^{2}=2px+p^{2}$.

A segment of length $l$ has its two endpoints sliding along the coordinate axes; find the envelope of the family of segments so produced.

Find the envelope of the family of ellipses $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}=1$, given that the area $S$ of each ellipse in the family is constant.

A shell is fired in a vacuum with initial speed $v_{0}$; as the angle of elevation $\alpha$ varies in the vertical plane, find the envelope of the shell's trajectories.

Prove that the envelope of the normals of a plane curve is the evolute of that curve.

Investigate the properties of the discriminant curve of each of the following families of curves ($c$ being the parameter):

Find the envelope surface of the family of spheres of radius $r$ with centers on the circle $x=R\cos t$, $y=R\sin t$, $z=0$ ($t$ being the parameter, $R>r$).

Find the envelope surface of the family of spheres

(where $\cos^{2}\alpha +\cos^{2}\beta +\cos^{2}\gamma =1$, and $t$ is the parameter).

Find the envelope surface of the family of ellipsoids $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}+\dfrac {z^{2}}{c^{2}}=1$, where the volume $V$ of each ellipsoid is constant.

Find the envelope surface of the family of spheres of radius $\rho$ with centers on the cone $x^{2}+y^{2}=z^{2}$.

A luminous point is located at the origin of coordinates. If $x_{0}^{2}+y_{0}^{2}+z_{0}^{2}>R^{2}$, find the shadow cone produced by the projection of the sphere

If the parameters $p$ and $q$ are subject to the equation

find the envelope surface of the family of planes

The equation $f(x,y,a)=0$, for various values of $a$, represents a family of curves. Sometimes this family has an envelope, i.e.a curve tangent to the curves of the family. The coordinates of the point of tangency of a curve of the family and the envelope satisfy the equations:

Eliminating the parameter $a$ from these, we obtain an equation satisfied by all points of the envelope. However, the same equations are also satisfied by the points of the locus of singular points of the curves of the family, i.e.points of the curves where $\dfrac {\partial f}{\partial x}=0$, $\dfrac {\partial f}{\partial y}=0$.

Find the envelope of the family of lines $(x-a)^{2}+y^{2}=1$.

Find the envelope of the family of lines $(x-a)^{2}+y^{2}=\dfrac {a^{2}}{2}$.

Find the envelope of the family of lines $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}=1$, where $a^{2}+b^{2}=1$.

Find the envelope of the family of lines $\lambda x^{2}+\mu y^{2}=1$, where $\lambda +\mu =\lambda \mu$.

Find the envelope of the family of lines $b^{2}x^{2}+a^{2}y^{2}=a^{2}b^{2}$, where $a+b=d$.

Find the envelope of the family of lines that cut off, from a given angle of magnitude $\omega$, a triangle of area $2m^{2}$.

Find the envelope of the family of lines $bx+ay=ab$, where $a+b=c$.

Find the envelope of the family of lines $bx+ay=ab$, where $a^{2}+b^{2}=c^{2}$.

The vertex of an angle of given magnitude $\alpha$ slides along the axis $Ox$, and one of its sides passes through the point $(0,h)$. Find the envelope of the other side.

A square with side $a$ moves so that two of its sides, $AB$ and $BC$, pass through the points $(\pm b, 0)$. Find the envelope of the diagonal $AC$.

A line rotates uniformly about a point moving with a given speed along a given line. Find the envelope of the moving line.

A circle rolls without slipping along a given line. Find the envelope of its diameter.

On the chords of the circle $x^{2}+y^{2}=a^{2}$ parallel to the axis $Oy$, circles are constructed having these chords as diameters. Find their envelope.

Find the envelope of the circles having their centers on a parabola and passing through its vertex.

Find the envelope of the circles constructed, as diameters, on the chords of a parabola that pass through the focus.

Find the envelope of the system of circles having their centers on a given ellipse and passing through the center of the ellipse.

Find the envelope of the circles with centers on the hyperbola $xy=a^{2}$ that are tangent to the axis $Ox$.

On the radius vectors of a curve, circles are constructed having them as diameters. Prove that the envelope of these circles is the pedal curve of the curve with respect to the pole. (The pedal curve of a curve with respect to a given point is the locus of the feet of perpendiculars dropped from the given point onto the tangents to the given curve; see problem 1981.)

The circle $x^{2}+y^{2}=r^{2}$ is the envelope of the family $\dfrac {x}{a}+\dfrac {y}{b}=1$. What is the relation between $a$ and $b$?

Find the envelope of the family of lines:

Find the envelope of the family of lines that form, with the coordinate axes, triangles of constant area $S$.

Find the envelope of the family of lines containing a segment of constant length $a$ whose endpoints slide along the coordinate axes.

Prove that the envelope of the normals of a plane curve is the evolute of that curve.

A bundle of parallel rays directed along the $x$-axis falls on the arc of the circle $x^{2}+y^{2} = a^{2}, x > 0$. Find the caustic, i.e., the envelope of the reflected rays.

Find the envelope of the family of circles:

Find the envelope of the family of circles having centers on the parabola $y^{2} = 2 x$ and passing through its vertex.

Find the envelope of the family of circles constructed on the focal chords of the parabola $y^{2} = 8 x$ as diameters.

Find the envelope of the family of ellipses $x^{2} / a^{2}+y^{2} / b^{2} = 1$, if the sum of the semi-axes of each ellipse is constant and equals $d$.

Find the envelope of the family of parabolas:

Find the parabola of safety, i.e., the envelope of the trajectories of projectiles launched with initial speed $v_{0}$ in a vertical plane at all possible launch angles $\alpha$ (air resistance neglected).

Find the envelope of the family of curves:

Find and investigate the discriminant curve of the family of curves:

Find the envelope of the family of surfaces:

Find the envelope of spheres of radius $R$ whose centers lie:

Find the envelope of the ellipsoids $x^{2} / a^{2}+y^{2} / b^{2}+z^{2} / c^{2} = 1$ having constant volume $V$.