Question

If `bb((15a^2 + 4b^2)/( 15a^2 - 4b^2) = 47/7)` then find the value of the following ratio. 

`[ b^2 - 2a^2]/[ b^2 + 2a^2]`

Answer 1

`(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^2 - 2a^2]/[ b^2 + 2a^2]`

Divided by `b^2`

=`[ b^2/b^2 - (2a^2)/b^2]/[ b^2/b^2 + (2a^2)/b^2]`

= `[ 1 - 2 xx 9/25]/[1 + 2 xx 9/25]`   ...`( a/b = 3/5 ⇒ a^2/b^2 = 9/25)`

= `[ (25 - 18)/25] /[( 25 + 18)/25] `

=`7/43`

Answer 2

`(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`   

∴ `a^2/b^2=(3)^2/(5)^2`  ...(Squaring both sides)

∴ `a^2/b^2=9/25`

∴ `a^2/b^2xx2=9/25xx2`    ...(Multiplying both sides by 2)

∴ `(2a^2)/b^2=18/25`

∴ `b^2/(2a^2)=25/18`    ...(using Invertendo)

∴ `(b^2+2a^2)/(b^2-2a^2)=(25+18)/(25-18)`   ...(using componendo-dividendo)

∴ `(b^2+2a^2)/(b^2-2a^2)=43/7`

∴ `(b^2+2a^2)/(b^2-2a^2)=7/43`   ...(using Invertendo)