Question

$\alpha ,\beta ,\lambda$ are the geometic means between ac,ab ; ab,bc ; bc,ca respectively and $a,b,c$ are in $A.P$. then ${\alpha }^2,{\beta }^2,{\lambda }^2$ are in

• A. A.P.
• B. G.P.
• C. H.P.
• D. A.G.P.

Answer 1

The correct option is A A.P.

As $\alpha ,\beta ,\lambda$ are the geometric menas bwtween ac,ab;ab,bc;bc,ca,

so ${\alpha }^2=a^{2}bc,{\beta }^2=b^{2}ac,{\lambda }^2=c^{2}ab$

Also a,b and c are in Ap

$2b=a+c$

Also

$\beta +\gamma ,\gamma +\alpha ,\alpha +\beta$ are in HP

$\therefore \frac{1}{\gamma +\alpha }-\frac{1}{\beta +\gamma }=\frac{1}{\alpha +\beta }-\frac{1}{\beta +\gamma }$

$\frac{\beta -\alpha }{\beta -\gamma }=\frac{\gamma -\beta }{\alpha +\beta }$

or ${\beta }^2-{\alpha }^2=-{\beta }^2+{\gamma }^2$

$⟹{\alpha }^2,{\beta }^{2}and{\gamma }^2$ are in AP