यदि `a=(sqrt(5)+1)/(sqrt(5)-1)` और `b=(sqrt(5)-1)/(sqrt(5)+1)` है तो `(a^(2)+ab+b^(2))/(a^(2)-ab+b^(2))` का मान ज्ञात करें?

Options

A. `3/4`
B. `4/3`
C. `3/5`
D. `5/3`

Answer

B

Explanation

`a=(sqrt(5)+1)/(sqrt(5)-1)b=(sqrt(5)-1)/(sqrt(5)+1)`
`:. a=1/b`
`a+b=a+1/a`
`implies (sqrt(5)+1)/(sqrt(5)-1)+(sqrt(5)-1)/(sqrt(5)-1)`
`implies (5+1+2sqrt(5)+5+1-2sqrt(5))/((sqrt(5))^(2)-(1)^(2))`
`implies 6+2sqrt(5)+6-2(sqrt(5))/(5-1)=12/4=3`
`:. (a^(2)+ab+b^(2))/(a^(2)-ab+b^(2))`
`=(a^(2)+1/(a^(2))+ab)/(a^(2)+1/(a^(2))-ab)`
`impliesa+1/a=3`
`a^(2)+1/(a^(2))=9-2=7 (ab=1)`
`:. (a^(2)+1/(a^(2))+ab)/(a^(2)+1/(a^(2))-ab)=(7+1)/(7-1)=8/6=4/3`