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प्रश्न
If (9a2 + 8b2) (9c2 − 8d2) = (9a2 − 8b2) (9c2 + 8d2), prove that a : b :: c : d.
सिद्धांत
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उत्तर
We have given that:
⇒ `(9a^2 + 8b^2)/(9c^2 - 8d^2) = (9a^2 - 8b^2)/(9c^2 + 8d^2)`
⇒ `(9a^2 + 8b^2)/(9a^2 - 8b^2) = (9c^2 + 8d^2)/(9c^2 - 8d^2)`
Apply the Componendo and Dividendo:
⇒ `((9a^2 + 8b^2) + (9a^2 - 8b^2))/((9a^2 + 8b^2) - (9a^2 - 8b^2)) = ((9c^2 + 8d^2) + (9c^2 - 8d^2))/((9c^2 + 8d^2) - (9c^2 - 8d^2))`
⇒ `(18a^2)/(16b^2) = (18c^2)/(16d^2)`
⇒ `(9a^2)/(8b^2) = (9c^2)/(8d^2)` ...(Dividing by 2)
⇒ `a^2/b^2 = c^2/d^2`
⇒ `sqrt(a^2/b^2) = sqrt(c^2/d^2)`
⇒ `a/b = c/d`
⇒ a : b :: c : d
Hence proved.
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