A calculus problem by Archit Tripathi

Assuming that $AB$ is on the line $y = 2x -17$ and the coordinates of the other two vertices on the parabola $C(x_{1},y_{1})$ and $D(x_{2},y_{2})$ . Then $CD$ is on a line whose equation is $y = 2x + b$ . Solving with parabola, we get $x^{2} = 2x + b$ .

$\Rightarrow$ $x_{1} = 1 + \sqrt{b + 1}$ , $x_{2} = 1 - \sqrt{b + 1}$

Assuming the length of the side of the square is $a$

$\Rightarrow$ $(x_{1} - x_{2})^{2} + (y_{1} - y_{2})^{2} = 5(x_{1} - x_{2})^{2} = 20(b + 1)$ ....(1)

Now, distance between the lines $y = 2x - 17$ and $y = 2x + b$ is $a$

$\Rightarrow$ $a = \frac{17 + b}{\sqrt{5}}, \frac{-(17 + b)}{\sqrt{5}}$ ...(2)

From (1) and (2), we get $b_{1} = 3, b_{2} = 63$ , so

$a^{2} = 80$ or $a^{2} = 1280$ $\Rightarrow$ $a^{2} = 80$ which is the required area, hence $[\sqrt{S}] = \boxed{8}$