The points \((1,3),(5,1)\) are opposite vertices of a diagonal of a rectangle. If the other two vertices lie on the line \(y=2 x+c\), then one of the vertex on the other diagonal is

Diagonals of rectangle bisect each other.
\(\therefore\) Midpoint of \((1,3)\) and \((5,1)\) is \((3,2)\).
Also, \(y=2 x+\) c passes through \((3,2)\).
\(\therefore 2 =2(3)+c\)
\(\therefore c =-4\)
\(\therefore \) Other two vertices lie on \(y=2 x-4\).
Let co-ordinates of \(\mathrm{B}\) be \((x, y)\) i.e., \((x, 2 x-4)\)
slope of \(\mathrm{AB} \times\) slope of \(\mathrm{BC}=-1\)
\(\begin{aligned}
& \Rightarrow\left(\frac{2 x-4-3}{x-1}\right)\left(\frac{2 x-4-1}{x-5}\right)=-1 \\
& \Rightarrow\left(\frac{2 x-7}{x-1}\right)\left(\frac{2 x-5}{x-5}\right)=-1 \\
& \Rightarrow x^2-6 x+8=0 \\
& \Rightarrow x=4,2
\end{aligned}\)
When \(x=4, y=4\)
When \(x=2, y=0\)
\(\therefore\) Vertex of the other diagonal is \((2,0)\).