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Question
What quantity must be added to each term of the ratio a + b: a - b to make it equal to (a + b)2 : (a - b)2 ?
Solution
Let the quantity to be added be x.
Then
`((a + b) + x)/((a - b) + x) = (a + b)^2/(a - b)^2`
⇒ (a + b) (a - b)2 + (a - b)2.x
= (a + b)2 (a - b) + (a + b)2.x
⇒ [(a + b)2 - (a - b)2]x
= (a2 - b2) (a - b) - (a2 - b2) (a + b)
⇒ (4abx) = (a2 - b2) [(a - b) - (a + b)]
⇒ x = `(-2b(a^2 - b^2))/(4ab) = (b^2 - a^2)/(2a)`.
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