Points on parabola with abscissa in A.P. and ordinate in G.P.

Question

The points with coordinates $(a,b),(a_1,b_1),(a_2,b_2)$ are points on parabola $y=3x^2$. The numbers $a,a_1,a_2$ are in Arithmetic progression while $b,b_1,b_2$ are in Geometric Progression. Calculate common ratio of geometric progression.

IS it possible to get a constant number (e.g. $1,2$ etc) as common ratio of G.P. because if we need answer in terms of given parameters, then answer would be $\frac{b_1}{b}$ or $\frac{b_2}{b_1}$

If I take three numbers $a,a+d,a+2d$ in A.P. and $b,br,br^2$ in G.P. and make three equations then I am not able to eliminate $d$. Could someone give me some hint as how to proceed.

Answer 1

WLOG let $a,a_1,a_2$ are $A,A\pm d$

$$\implies3(A+D)^2\cdot3(A-D)^2=\{3A^2\}^2$$

$$\iff A^4=(A^2-D^2)^2\iff D^4-2A^2D^2=0$$

$$\implies D^2=2A^2$$ as $D\ne0$

We need $$\dfrac{A+d}A=\dfrac A{A-d}=?$$