The area of a rectangle and the square of its perimeter are in the ratio 1 ∶ 25. Then the lengths of the shorter and longer sides of the rectangle are in the ratio

Let 'a' and 'b' be the length of sides of the rectangle. (a > b)

Area of the rectangle = a*b

Perimeter of the rectangle = 2*(a+b)

$$\Rightarrow$$ $$\dfrac{a*b}{(2*(a+b))^2}=\dfrac{1}{25}$$

$$\Rightarrow$$ $$25ab=4(a+b)^2$$

$$\Rightarrow$$ $$4a^2-17ab+4b^2=0$$

$$\Rightarrow$$ $$(4a-b)(a-4b)=0$$

$$\Rightarrow$$ $$a = 4b$$ or $$\dfrac{b}{4}$$

We initially assumed that a > b, therefore a $$\neq$$ $$\dfrac{b}{4}$$.

Hence, a = 4b

$$\Rightarrow$$ b : a = 1 : 4