If the number x,y,z are in H.P. , then `sqrt(yz)/(sqrt(y)+sqrt(z)),sqrt(xz)/(sqrt(x)+sqrt(z)),sqrt(xy)/(sqrt(x)+sqrt(y))` are in

Options

A. AP
B. GP
C. HP
D. None of these

Answer

A

Explanation

`:.x,y,z` are in HP.
`:.(1)/(x),(1)/(y),(1)/(z)` are in AP.
`:.(1)/(x)-(1)/(y)=(1)/(y)-(1)/(z)" " ".......(i)"`
`(sqrtyz)/(sqrt(y)+sqrt(z))=(1)/((1)/(sqrt(y))+(1)/(sqrt(z)))=a" " [" say "]`
`(sqrtzx)/(sqrt(x)+sqrt(z))=(1)/((1)/(sqrt(z))+(1)/(sqrt(x)))=b" " [" say "]`
and `(sqrtxy)/(sqrt(x)+sqrt(y))=(1)/((1)/(sqrt(x))+(1)/(sqrt(y)))=c" " [" say "]`
`:.(a-b)/(b-c)=((1)/((1)/(sqrt(y))+(1)/(sqrt(z)))-(1)/((1)/(sqrt(z))+(1)/(sqrt(x))))/((1)/((1)/(sqrt(z))+(1)/(sqrt(x)))-(1)/((1)/(sqrt(x))+(1)/(sqrt(y))))=((1)/(x)-(1)/(y))/((1)/(y)-(1)/(z))=(a)/(a)`
`" " " " [" from Eq. (i) "]`
Hence, `(sqrtyz)/(sqrt(y)+sqrt(z)),(sqrt(zx))/(sqrt(z)+sqrt(x)),(sqrt(xy))/(sqrt(x)+sqrt(z))` are in AP.