Lemniscate, Cassini and Descartes ovals

Find the radius of curvature of the curve given by the following polar equation: the lemniscate $r^{2}=a^{2}\cos 2\varphi$.

Find the locus of points the product of whose distances to two given points $(a,0)$ and $(-a,0)$ is a constant quantity equal to $b^{2}$. (Ovals of Cassini.)

Find the locus of points the product of whose distances to two given points equals the square of half the distance between them. (Lemniscate of Bernoulli.)

Find the locus of points between whose distances to two given points there holds the linear relation $r_{1}-ar_{2}=b$. (Ovals of Descartes.)

Find the angle between the tangent and the polar radius of the point of tangency for the following curves:

Plot the graph of the equation by passing to polar coordinates:

Find the area of the figure bounded by curves given in polar coordinates.

Find the volume of the solid formed by rotating one loop of the lemniscate $r^{2} = a^{2} \sin 2 \varphi$:

Find the volume of the solid formed by rotating around the line $y = x$ the figure bounded by the curve $\left(x^{2}+y^{2}\right)^{2} = a^{2}\left(x^{2}-y^{2}\right)$.

Given $n$ points $P_{1},P_{2},\ldots ,P_{n}$ in a plane in which $P$ varies. The locus of the points $P$ for which the product of the distances

is a ``lemniscate with $n$ foci''. (The standard lemniscate represents the special case $n=2$, cf.problem 4.) Show that a lemniscate with $n$ foci can never consist of more than $n$ separate closed branches.

(Lemniscate of Bernoulli.) Two points $F_{1}$ and $F_{2}$ are given, at a distance $2 c$ from each other. Find the equation of the locus of points for which the product of the distances to the points $F_{1}$ and $F_{2}$ equals $c^{2}$, taking as the origin of coordinates the midpoint of the segment $F_{1} F_{2}$, and as the abscissa axis the line $F_{1} F_{2}$. Pass to the corresponding polar coordinate system. Sketch the curve.

At what angle does the lemniscate of Bernoulli from the preceding problem intersect the axis $O x$:

(Oval of Descartes.) In a polar coordinate system, find the equation of the locus of points whose distances $\rho_{A}$ and $\rho_{B}$ to two fixed points $A$ and $B$, at a distance $c$ from each other, satisfy the equality $\rho_{B} = a \rho_{A}+b$, taking $A$ as the pole, and the line $A B$ as the polar axis.

In right triangles formed by the coordinate axes and the lines intersecting them, lying in the first and third quadrants and having one and the same area $s > 0$, perpendiculars are dropped from the origin onto the hypotenuse. Find the locus of the feet of these perpendiculars. Sketch it.