Question

Let $a,x,b$ be in A.P.; $a,y,b$ be in G.P. and $a,z,b$ be in H.P. if $x=y+2$ and $a=5z$ then

• A. $y^2=xz$
• B. $x>y>z$
• C. $a=9,b=1$
• D. $a=\frac{9}{4},b=\frac{1}{4}$

Answer 1

Answer: (A), (B), (C)

The correct options are
A $y^2=xz$
B $x>y>z$
C $a=9,b=1$

$2x=a+b$ .... (i)

$y^2=ab$ ... (ii)

$z=\frac{2ab}{a+b}$ .... (iii)

from (i), (ii) and (iii) $y^2=xz$

Also $a=5z\Rightarrow a=5\left(\frac{2ab}{a+b}\right)\Rightarrow a+b=10b$

$\Rightarrow a=9b$

Also $x=y+2\Rightarrow \frac{a+b}{2}=\sqrt{ab}+2$

but $a=9b$

$\therefore 5b=\sqrt{9b^2}+2\Rightarrow 5b=3b+2$

$\therefore b=1\Rightarrow a=9$

also AM>GM>HM $\Rightarrow x>y>z$