Is the boundary a circle?

Let $u=3x+2y$ and $v=5x-ky$ . Then the Jacobian of $u$ and $v$ with respect to $x$ and $y$ is equal $J=\begin{vmatrix}3&2\\5&{-k}\end{vmatrix}=-3k-10.$ As $k$ is a positive number then $|J|=3k+10.$ Let us represent the region under consideration by $R_{x,y},$ and its transformed region in terms of the coordinates $u$ and $v$ as $R_{u, v}$ . Obviously, $R_{u, v},$ is the circle with center at $(0, 0)$ and radius 5. Then $25 \pi=area(R_{u, v})=\iint_{R_{u, v}}du dv=\iint_{R_{x, y}}(3k+10)dx dy=(3k+10) area(R_{x,y}.)$ Solving this equation for $area(R_{x,y}),$ we obtain that $area( R_{x,y})= \frac{25 \pi}{3k+10},$ therefore, $\frac{25 \pi}{3k+10}=\frac{25 \pi}{22}.$ Hence the answer to this question is $k=\boxed{4.00}.$