Question

If $a,b,c$ are in A.P., $a,x,b$ are in GP and $b,y,c$ are also in G.P, then $x^2,b^2,y^2$ are in AP.

• A. True
• B. False

Answer 1

The correct option is A

True

Given $a,b,c$ are in A.P..

$\Rightarrow 2b=a+c...\left(1\right)$

Given $a,x,b$ are in G.P..

$\Rightarrow x^2=ab...\left(2\right)$

Also, given $b,y,c$ are in G.P..

$\Rightarrow y^2=bc...\left(3\right)$

Adding (2) and (3), we get

$x^2+y^2=ab+bc.$

i.e., $x^2+y^2=b(a+c)$

$=b\left(2b\right)[\text{Using (1)]}$

$=2b^2$

i.e., $2b^2=x^2+y^2$

$\therefore x^2,b^2,y^2$ are in A.P..