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Question 12 : 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

3 : 8
2 : 9
1 : 4
1 : 3

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Explanatory Answer

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Explanatory Answer

Method of solving this CAT Question from Mensuration

Given that the area of a rectangle and the square of its perimeter are in the ratio 1: 25, we have to find the ratio of the lengths of the shorter and longer sides of the rectangle i.e. b : l
The area of the rectangle = lb
The square of its perimeter = [2(l + b)]²
⟹ lb : [2(l + b)]² = 1 : 25
⟹ lb : 4[l² + 2lb + b²] = 1 : 25
⟹ 25 lb = 4[l² + 2lb + b²]
⟹ 25 lb = 4l² + 8lb + 4b²
⟹ 4l² - 17lb + 4b² = 0
⟹ 4l² - 16lb - lb + 4b² = 0
⟹ 4l[l- 4b] – b [l- 4b] = 0
⟹ [l - 4b][4l - b] = 0
By this we can find that the ratio of the shorter and longer sides of the rectangle b : l is in the ratio 1: 4

In other way ,if we want to deal with only one variable and do not want to factorize divide this 4l² - 17lb + 4b² = 0 by l² throughout
⟹ \\frac{4l^2}{l^2}) - \\frac{17lb}{l^2}) + \\frac{4b^2}{l^2})
⟹ 4 - \\frac{17b}{l}) + 4[\\frac{b}{l})]²
Since we have to find ratio of b : l, \\frac{b}{l}) can be substituted with x
4 - 17x + 4x² = 0
Now x can be found by simplifying it

The question is "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"