Question : The square of the difference between two given natural numbers is 324, while the product of these two given numbers is 144. Find the positive difference between the squares of these two given numbers.
Option 1: 630
Option 2: 540
Option 3: 450
Option 4: 360
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Correct Answer: 540
Solution :
Let $a$ and $b$ be the two numbers.
So, $ab$ = 144 and $(a-b)^2 = 324$
⇒ $ab$ = 144 and $(a-b) = 18 $
We know that $(a+b)^2 = (a-b)^2 + 4ab$
⇒ $(a+b)^2 = 324 + 576$
⇒ $(a+b)^2 = 900$
⇒ $(a+b) = 30$
Now, $a^2-b^2$ = $(a-b)×(a-b)$
⇒ $a^2-b^2 = 18 × 30$
$\therefore a^2-b^2 = 540$
Hence, the correct answer is 540.
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