If `a ,ba n dc` are in A.P., and `pa n dp '` are respectively, A.M. and G.M. between `aa n dbw h i l eq , q '` are , respectively, the A.M. and G.M. between `ba n dc ,` then `p^2+q^2=p^('2)+q^('2)` b. `p q=p ' q '` c. `p^2-q^2=p^('2)-q^('2)` d. none of these

Options

A. `p^2+q^2=P'^2+q'^2`
B. `pq=p'q'`
C. `p^2-q^2=p'^2-q'^2`
D. none of these

Answer

C

Explanation

2b=a+c
a,p,b,q,c are in A.P. Hence,
`p=(a+b)/2 and q=(b+c)/2`
Again a,p',b,q' and c are in G.P are in G.P. Hence,
`p'=sqrt(ab)andq'=sqrt(bc)`
`rArrp^(2)-q^(2)=((a-c)(a+c+2b))/4`
=(a-c)b
=ab-bc
=`p'^(2)-q'^(2)`