If $a, b, c$ are in $AP$ and $a^2,b^2,c^2$ are in $HP$, how are $a, b$ and $\large\frac{-c}{2}$ related?

Question

$\begin{array}{1 1} Not\;related \\a, b, \frac{-c}{2}\; are \infty \\a, b, \frac{-c}{2} are\; in\; AP \\ a, b, \frac{-c}{2}\; are\; in\; GP \end{array}$

Answer 1

Answer : (d) $a,b,\frac{-c}{2}\;are\;in\;GP$

Explanation :$a,b,c\;are\;in\;AP$

$2b=a+c\qquad\;b^2=\frac{(a+c)^2}{4}$

$a^2,b^2,c^2\;are\;in\;HP$

$b^2=\frac{2a^2c^2}{a^2+c^2}$

$b^2=\frac{2a^2c^2}{4b^2-2ac}$

$(ac-b^2)(ac+2b^2)=0$

$b^2=ac\qquad\;or\qquad\;-2b^2=ac$

$b^2=ac\;along\;with\;\frac{a+c}{2}=b$

$a=b=c.$

$-2b^2=ac\qquad\;a,b,\frac{-c}{2}\;are\;in\;GP.$