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Question
The sum of the squares of two positive integers is 208. If the square of the large number is 18 times the smaller. Find the numbers.
Solution
Let the two numbers be x and y, y being the bigger number.
From the given information,
x2 + y2 = 208 ...(i)
y2 = 18x ...(ii)
From (i), we get y2 = 208 – x2
Putting this in (ii), we get,
208 – x2 = 18x
⇒ x2 + 18x – 208 = 0
⇒ x2 + 26x – 8x – 208 = 0
⇒ x(x + 26) – 8(x + 26) = 0
⇒ (x – 8)(x + 26) = 0
⇒ x can't be a negative number, hence x = 8
⇒ Putting x = 8 in (ii), we get y2 = 18 × 8 = 144
⇒ y = 12, since y is a positive integer
Hence, the two numbers are 8 and 12.
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