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Question
If `bb((15a^2 + 4b^2)/( 15a^2 - 4b^2) = 47/7)` then find the value of following ratio.
`[ b^3 - 2a^3]/[ b^3 + 2a^3]`
Solution
`(15a^2 + 4b^2)/( 15a^2 - 4b^2) = 47/7`
Applying componendo and dividendo, we get
`[(15a^2 + 4b^2) + ( 15a^2 - 4b^2)]/[(15a^2 + 4b^2) - ( 15a^2 - 4b^2)] = [47 + 7]/[47-7]`
⇒ `[15a^2 + 4b^2 +15a^2 - 4b^2]/[15a^2 + 4b^2 - 15a^2 + 4b^2]` =`54/40`
⇒ `(30a^2)/(8b^2) = 27/20`
⇒ `a^2/b^2 = (27 xx 8)/( 20 xx 30) = 9/25`
⇒ `a/b = sqrt( 9/25) = 3/5`
` therefore [ b^3 - 2a^3]/[ b^3 + 2a^3]`
Divided by `b^3`
=`[ b^3/b^3 - (2a^3)/b^3]/[ b^3/b^3 + (2a^3)/b^3]`
= `[ 1 - 2 xx 27/125]/[1 + 2 xx 27/125]` ...`( a/b = 3/5 ⇒ a^3/b^3 = 27/125)`
= `[ (125 - 54)/125] /[(125 + 54)/125] `
=`71/179`
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