Question

# If the A.M. and G.M. between two numbers are in ratio of m:n, then the ratio of numbers can be m+√m2−n2m−√m2−n2m+√m2−n2n−√m2−n2 m+√m2+n2m−√m2+n2n+√m2−n2m−√m2−n2

Solution

## The correct option is A m+√m2−n2m−√m2−n2Let the two numbers be a and b, so A=a+b2,  G=√ab⇒a+b=2A,  ab=G2⋯(1) The equation whose roots are a and b is x2−(a+b)x+ab=0⇒x2−2Ax+G2=0⇒x=2A±√4A2−4G22⇒x=A±√A2−G2 So, the two numbers are a=A+√A2−G2 and b=A−√A2−G2 It is given that A:G=m:n⇒A=km,G=kn Substituting the values of A and G, we get ab=km+√k2m2−k2n2km−√k2m2−k2n2∴ab=m+√m2−n2m−√m2−n2

Suggest corrections