Find the maximum and minimum radii vectors of section of the surface $(x^2+y^2+z^2)^2=a^2x^2+b^2y^2+c^2z^2$ by the plane $lx+my+nz=0$

Question

Find the maximum and minimum radii vectors of section of the surface.

$(x^2+y^2+z^2)^2=a^2x^2+b^2y^2+c^2z^2$ by the plane $lx+my+nz=0$

I formed an expression $F=x^2+y^2+z^2 + \alpha((x^2+y^2+z^2)^2-a^2x^2+b^2y^2+c^2z^2) + \beta(lx+my+nz).$ I partially differentiated the expression wrt $x,y$ and $z$, and equated it to $0$ to obtain the stationary points. I obtained $\alpha=-\frac{1}{r^2}$, and $\beta=\frac{2a^2lx+2b^2my+2c^2nz}{-r^2(l^2+m^2+n^2)}$ where $r=\sqrt{(x^2+y^2+z^2)}$.

I also found $x=\frac{-\beta lr^2}{2(a^2-r^2)}, y=\frac{-\beta mr^2}{2(b^2-r^2)}$ and $z=\frac{-\beta nr^2}{2(c^2-r^2)}$. On substituting the value of $\beta$ the expression for $x,y$ and $z$ gets even more complex. I am not able to find $x,y$ and $z$ individually.

If anyone can help me with a solution for this by Lagrange's multiplier method, I would be really happy.

Answer 1

I expect we can solve this problem using its geometrical interpretation.

As the radius ($r(x,y,z)$) we’ll be understand a distance from a point $(x,y,z)$ ofm the section to the origin $(0,0,0)$. Then the minimum radius is zero, because the section contains the origin.

To find the maximum radius we’ll proceed as follows. The left hand side of the surface equation is $r^4$. With this value fixed the right hand side of the equality is an equation of an ellipsoid (provided all $a,b,c$ are non-zero). Its intersection with the plane given by the equation $lx+my+nz=0$ is an ellipse (this intersection should be an ellipse even if one of $a,b,c$ is zero and I’ll skip the remaining degenerate cases). Now the equality between the left hand side and the right hand side of the equation describes the intersection of the ellipse with the circle of radius $r^2$ centered at the origin. Since when $r$ tends to the infinity, the left hand side describes the family of circles homothetic with the coefficient $r^2$ and the center at the origin, whereas the right hand describes the family of ellipses homothetic with the coefficient $r$ and the same center, eventually the intersection will be empty, and the maximum $r$ for which it is non-empty is attained when the endpoints of the big axis of the ellipse are exactly at the distance $r^2$ from the origin. This is a geometric condition for the maximum radius of the section, which should be converted to an analytical one.

I am stuck at this problem since 2 days.

Your are too hasty. I have problems at which I am stuck from the former millennium. :-)