Question

Three numbers $a,b,c$ are in G.P. If $4a,5b$ and $4c$ are in A.P. and $a+b+c=70,$ then $|c-a|$ equals

• A. $10$
• B. $20$
• C. $30$
• D. $40$

Answer 1

The correct option is C $30$

Since $a,b,c$ are in G.P.,
$\therefore b^2=ac\dots \left(1\right)$
$4a,5b,4c$ are in A.P.
$\therefore 10b=4a+4c\dots \left(2\right)$
and $a+b+c=70\dots \left(3\right)$

From $\left(2\right)$ and $\left(3\right),$
$10b=4(70-b)$
$\Rightarrow b=20$

From $\left(1\right)$ and $\left(3\right),$
$ac={20}^2$ and $a+c=50$
$|c-a|=\sqrt{(a+c{)}^2-4ac}$
$=\sqrt{{50}^2-4\times {20}^2}=30$