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 $a=9,b=1$

$x,y$ and $z$ are $AM,GM$ and $HM$ between $a,b$ respectively.
$\Rightarrow x=\frac{a+b}{2}$ ----(1)
$\Rightarrow y=\sqrt{ab}$ ------(2)
$\Rightarrow z=\frac{2ab}{a+b}$ ------(3)
But, given $a=5z$ and $x=y+2$
Substituting $a=5z$ in (3) gives $a=9b$
Substituting $a=9b$ in $\left(1\right)$ and $\left(2\right)$
$\Rightarrow x=5b$ and $y=3b$
$x=y+2\Rightarrow b=1$

Also, we get $z=\frac{9}{5}b$

Since, $AM\ge GM\ge HM$ and $G{M}^2=\left(AM\right)\left(HM\right)$
$\therefore x\ge y\ge z$ and $y^2=xz$ and $a=9,b=1$
Hence, option A,B and C are correct.