UPSC NDA Math Model Paper 10

Let the numbers be a and b
According to given condition,

a+b

2√ab

=

m

n

Applying componendo and dividendo, we get

a+b+2√ab

a+b−2√ab

=

m+n

m−n

(√a+√b)2

(√a−√b)2

=

m+n

m−n

(√a+√b)

(√a−√b)

=

√m+n

√m−n

Applying componendo and dividendo again, we get

(√a+√b)+(√a−√b)

(√a+√b)−(√a−√b)

=

√m+n+√m−n

√m+n−√m−n

Squaring both sides, we get

a

b

=(

√m+n+√m−n

√m+n−√m−n

)2

a

b

=

2m+√m2−n2

2m−2√m2−n2

a

b

=

m+√m2−n2

m−√m2−n2

Hence, option (1) is correct.