Question : The sum of three positive numbers is 18 and their product is 162. If the sum of two numbers is equal to the third number, then the sum of the squares of the numbers is:
Option 1: 120
Option 2: 126
Option 3: 132
Option 4: 138
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Correct Answer: 126
Solution :
Let three numbers be $a, b$ and $c$.
Given: $a + b + c = 18, a + b = c$ and $abc = 162$
Put $a + b = c$ in $a + b + c = 18$
$2c = 18$
⇒ $c = 9 = a + b$ ---(1)
$ab × 9 = 162$
⇒ $ab = \frac{162}{9}$
⇒ $ab = 18 $ ---(2)
From (1) and (2) we get,
⇒ $a=6, b=3$
So, the sum of the squares of the numbers
= $(a^{2}+b^{2}+c^{2}) = (6^{2}+3^{2}+9^{2})= (36+9+81)= 126$
Hence, the correct answer is 126.
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