Question : If $a, b$ and $c$ are positive numbers such that $(a^2+b^2):(b^2+c^2):(c^2+a^2)=34:61:45$, then $b - a : c - b : c - a$ = _______.
Option 1: 3 : 2 : 1
Option 2: 2 : 1 : 3
Option 3: 3 : 1 : 2
Option 4: 1 : 2 : 3
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Correct Answer: 2 : 1 : 3
Solution :
Given that $(a^2+b^2):(b^2+c^2):(c^2+a^2)=34:61:45$
1. $a^2 + b^2 = 34k$ (for some constant $k$)
2. $b^2 + c^2 = 61k$
3. $c^2 + a^2 = 45k$
Adding these three equations, we get,
$⇒2(a^2 + b^2 + c^2) = 140k$
$\therefore a^2 + b^2 + c^2 = 70k$
Subtract the first equation from this result, we get,
$⇒c^2 = 36k$
$\therefore c = 6\sqrt{k}$
Similarly, by subtracting the second equation from the result, we get,
$⇒a^2 = 9k$
$\therefore a = 3\sqrt{k}$
And subtracting the third equation from the result, we get,
$⇒b^2 = 25k$
$\therefore b = 5\sqrt{k}$
$⇒(b - a) = 5\sqrt{k} - 3\sqrt{k} = 2\sqrt{k}$
$⇒(c - b) = 6\sqrt{k} - 5\sqrt{k} = \sqrt{k}$
$⇒(c - a) = 6\sqrt{k} - 3\sqrt{k} = 3\sqrt{k}$
$\therefore b - a : c - b : c - a = 2\sqrt{k} : \sqrt{k} : 3\sqrt{k}= 2 : 1 : 3$
Hence, the correct answer is 2 : 1 : 3.
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