Given information:
- The area of a rectangle and the square of its perimeter are in the ratio 1:25.

To find:
The ratio between the lengths of the shorter and longer sides of the rectangle.

Solution:

Let the length and width of the rectangle be 'l' and 'w' respectively.

Step 1: Formulating the equation
- Area of the rectangle = length × width = lw
- Perimeter of the rectangle = 2(length + width) = 2(l + w)

Given that the area of the rectangle and the square of its perimeter are in the ratio 1:25, we can write the equation as:

lw : (2(l + w))^2 = 1 : 25

Simplifying the equation, we get:

lw : 4(l^2 + 2lw + w^2) = 1 : 25

Step 2: Simplifying the equation
- Cross-multiplying, we have:
25lw = 4(l^2 + 2lw + w^2)

- Expanding, we get:
25lw = 4l^2 + 8lw + 4w^2

- Rearranging the equation, we have:
4l^2 - 17lw + 4w^2 = 0

Step 3: Solving the quadratic equation
- We can solve this quadratic equation by factoring or using the quadratic formula. For simplicity, let's use factoring.

- Factoring the equation, we get:
(2l - w)(2w - l) = 0

- Setting each factor equal to zero and solving, we have two possibilities:
1) 2l - w = 0 => 2l = w => l/w = 1/2
2) 2w - l = 0 => 2w = l => w/l = 1/2

Step 4: Finding the ratio of the shorter and longer sides
From the possibilities above, we can see that the ratio of the shorter side (w) to the longer side (l) is 1:2 (when l/w = 1/2) or 2:1 (when w/l = 1/2).

However, we need to find the ratio of the shorter and longer sides, not vice versa. Therefore, we take the reciprocal of each ratio to get the desired ratio:

1:2 => 2:1
or
2:1 => 1:2

Hence, the lengths of the shorter and longer sides of the rectangle are in the ratio 1:2 or 2:1.

Conclusion:
The correct answer is '1:2' or '2:1'.