Problems on Trajectories

Find the orthogonal trajectories of the family of parabolas $y=ax^{2}$.

Find the orthogonal trajectories of the family of curves $y=ax^{\sigma }$ for a given $\sigma$.

Find the orthogonal trajectories of the family of ellipses $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}=\lambda$ for given $a$ and $b$.

Find the orthogonal trajectories of the family of ellipses $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}=1$ for a given $a$.

Find the orthogonal trajectories of the family of curves $x^{\sigma }+y^{\sigma }=a^{\sigma }$.

Find the orthogonal trajectories of the family of confocal ellipses $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{a^{2}-c^{2}}=1$ for a given $c$.

Find the orthogonal trajectories of the family of circles $(x-\lambda )^{2}+y^{2}=a^{2}$ for a given $a$.

Find the orthogonal trajectories of the family of parabolas $(y-h)^{2}=2px$, sliding without rotation with their vertex along the axis $Oy$.

Find the orthogonal trajectories of the family of parabolas $(y-\eta )^{2}=2p(x-\xi )$, where $\eta^{2}+2p\xi =0$, moving without rotation while passing through the origin.

Find the orthogonal trajectories of the family of cissoids $(2a-x)y^{2}=x^{3}$.

Find the orthogonal trajectories of the family of lemniscates $(x^{2}+y^{2})^{2}=a^{2}(x^{2}-y^{2})$.

Find the orthogonal trajectories of the family of strophoids $x(x^{2}+y^{2})=a(x^{2}-y^{2})$.

Find the orthogonal trajectories of the family of parabolas for which $Ox$ is the axis of symmetry and $Oy$ is the directrix.

Find the orthogonal trajectories of the family of circles tangent to the two lines $y=\pm ax$ for a given $a$.

Find the orthogonal trajectories of the family of cardioids $r=a(1+\cos \varphi )$.

Find the orthogonal trajectories of the family of curves $r^{2}(\pi -\varphi )=a^{2}\varphi$.

Find the orthogonal trajectories of the family of curves $r^{2}=\ln \tg \varphi +C$.

Find the orthogonal trajectories of the family of curves $r^{2n}-2a^{n}r^{n}\cos n\varphi +a^{2n}=c^{2n}$ for given $a$ and $n$.

Find the involute of the parabola, i.e., the orthogonal trajectory of its tangents.

Find the involute of the catenary $y=a\ch \dfrac {x}{a}$.

Find the involute of the involute of a circle

Find the curves that intersect the cardioids $r=a(1+\cos \varphi )$, for any $a$, at an angle $\alpha$.

The same (find the curves that intersect the family, for any $a$, at an angle $\alpha$) for the curves $r^{2}\cos 2\varphi =a^{2}$.

The same (find the curves that intersect the family, for any $a$, at an angle $\alpha$) for the circles $r=a\cos \varphi$.

Find the family of curves $r=af(\varphi )$ such that each of the curves, when rotated through any angle $\beta$, intersects all the other curves of the family at an angle $\alpha$, where $\alpha$ is some function of $\beta$.

Find the curves that intersect at an angle of $45°$ the tangents to the circle $x^{2}+y^{2}=a^{2}$.

Find the curves that intersect all generators of the cone $\dfrac {y}{z}=\varphi \left(\frac{x}{z}\right)$ at a constant angle $\alpha$.

Find the line intersecting at an angle $\alpha$ the meridians of a paraboloid of revolution.

The same question (find the line intersecting the meridians at an angle $\alpha$) for the torus

Find the surface of revolution about $Oz$ such that the projections onto the plane $xOy$ of the lines on the surface that intersect the meridians at an angle $\alpha$ have the form of parabolas with focus at the origin.

Find the lines of steepest descent of the ellipsoid surface

Find the lines of steepest descent of the oblique circular cylinder $x^{2}+(y-z\tg \alpha )^{2}=a^{2}$.

Find the lines of steepest descent of the right circular cylinder tilted from the vertical by an angle $\alpha$: $x^{2}+(y\cos \alpha -z\sin \alpha )^{2}=a^{2}$.

The same question as in the previous problem (find the lines of steepest descent) for the oblique parabolic cylinder $y^{2}=2p(x-z\tg \alpha )$.