Task 301

The task

For $0 < k < 6,$ the graphs of $\frac{(x - k)^2}{9} + y^2 = 1$ and $\frac{x^2}{9} + y^2 = 1$ intersect at $A$ and $C,$ and have $x$-intercepts at $B$ and $D$ respectively. Compute the value of $k$ for which $ABCD$ is a square. [asy] unitsize(1 cm); path ellone = xscale(3)*Circle((0,0),1); path elltwo = shift((24/5,0))*xscale(3)*Circle((0,0),1); pair A, B, C, D; A = intersectionpoints(ellone,elltwo)[0]; C = intersectionpoints(ellone,elltwo)[1]; B = (-3 + 24/5,0); D = (3,0); draw(ellone); draw(elltwo); draw((-3.5,0)--(9,0)); draw((0,-1.5)--(0,1.5)); draw(A--B--C--D--cycle); label("$A$", A, N, fontsize(10)); label("$B$", B, NW, fontsize(10)); label("$C$", C, S, fontsize(10)); label("$D$", D, NE, fontsize(10)); [/asy]