Question

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

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

Answer 1

Answer: (A)

A.P.

By given condition

${\alpha }^2=a^{2}bc,{\beta }^2=b^{2}ca,{\gamma }^2=c^{2}ab$ and $2b=a+c$

${\alpha }^2,{\beta }^2,{\gamma }^2$ will him A.P if $a^{2}bc,b^{2}ca,c^{2}ab$ are in A.P or $a,b,c$ are in $A.P$ which is true

Again $\beta +\gamma ,\gamma +\alpha ,\alpha +\beta$ will be in $H.P$ is

$\frac{1}{\beta +\gamma },\frac{1}{\gamma +\alpha },\frac{1}{\alpha +\beta }$ are in $A.P$

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

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

or ${\alpha }^2,{\beta }^2,{\gamma }^2$ are in $A.P$