Given equation, x² - 6x - 2 = 0
Roots are $$\alpha $$ and $$\beta $$.
So, $$\alpha + \beta = 6$$ and $$\alpha \beta = - 2$$
In the question given, $${a_n} = {\alpha ^n} - {\beta ^n}$$
$$\therefore$$ $${a_8} = {\alpha ^8} - {\beta ^8}$$
and $${a_9} = {\alpha ^9} - {\beta ^9}$$
and $${a_{{10}}} = {\alpha ^{{10}}} - {\beta ^{10}}$$
Now, the given equation
$${{{a_{10}} - 2{a_8}} \over {2{a_9}}}$$
= $${{{\alpha ^{10}} - {\beta ^{10}} - 2\left( {{\alpha ^8} - {\beta ^8}} \right)} \over {2\left( {{\alpha ^9} - {\beta ^9}} \right)}}$$
=$${{{\alpha ^{10}} - {\beta ^{10}} + \alpha \beta \left( {{\alpha ^8} - {\beta ^8}} \right)} \over {2\left( {{\alpha ^9} - {\beta ^9}} \right)}}$$ (as $$\alpha \beta = - 2$$)
=$${{{\alpha ^{10}} - {\beta ^{10}} + {\alpha ^9}\beta - \alpha {\beta ^9}} \over {2\left( {{\alpha ^9} - {\beta ^9}} \right)}}$$
= $${{{\alpha ^9}\left( {\alpha + \beta } \right) - {\beta ^9}\left( {\alpha + \beta } \right)} \over {2\left( {{\alpha ^9} - {\beta ^9}} \right)}}$$
= $${{\left( {\alpha + \beta } \right)\left( {{\alpha ^9} - {\beta ^9}} \right)} \over {2\left( {{\alpha ^9} - {\beta ^9}} \right)}}$$
= $${{\left( {\alpha + \beta } \right)} \over 2}$$
= $${6 \over 2}$$ (as $${ {\alpha + \beta } }$$ = 6)
= 3

Let $$p,q,r$$ are in $$AP$$

$$ \Rightarrow 2q = p + r\,\,\,\,\,\,\,\,\,\,\,\,...\left( i \right)$$

Given $${1 \over \alpha } + {1 \over \beta } = 4 \Rightarrow {{\alpha + \beta } \over {\alpha \beta }} = 4$$

We have $$\alpha + \beta = - q/p$$ and $$\alpha \beta = {r \over p}$$

$$ \Rightarrow {{ - {q \over p}} \over {{r \over p}}} = 4 \Rightarrow q = - 4r\,\,\,\,\,\,\,\,...\left( {ii} \right)$$

From $$(i),$$ we have

$$2\left( { - 4r} \right) = p + r \Rightarrow p = - 9r$$

$$q = - 4r$$

Now $$\left| {\alpha - \beta } \right| = \sqrt {{{\left( {\alpha + \beta } \right)}^2} - 4\alpha \beta } $$

$$ = \sqrt {{{\left( {{{ - q} \over p}} \right)}^2} - {{4r} \over p}} $$

$$ = {{\sqrt {{q^2} - 4pr} } \over {\left| p \right|}}$$

$$ = {{\sqrt {16{r^2} + 36{r^2}} } \over {\left| { - 9r} \right|}}$$

$$ = {{2\sqrt {13} } \over 9}$$

Given, $$ - 3{\left( {x - \left[ x \right]} \right)^2} + 2\left( {x - \left[ x \right]} \right) + {a^2} = 0$$
As we know, $$\left[ x \right] + \left\{ x \right\} = x$$
where $$\left[ x \right]$$ is integral part and $$\left\{ x \right\}$$ is fractional part.
$$\therefore$$$$\left\{ x \right\} = x - \left[ x \right]$$
Now put $$\left\{ x \right\}$$ inplace of $$x - \left[ x \right]$$ in the equation.
The new equation is $$ - 3{\left\{ x \right\}^2} + 2\left\{ x \right\} + {a^2} = 0$$

[Note : Question says this equation has no integral solution, it means $$\left\{ x \right\} \ne $$ 0. So, $$x$$ is not a integer.]

$$\therefore$$ $$\left\{ x \right\}$$ = $${{ - 2 \pm \sqrt {4 - 4 \times \left( { - 3} \right){a^2}} } \over { - 6}}$$
             = $${{ - 2 \pm \sqrt {4 + 12{a^2}} } \over { - 6}}$$
As $$\left\{ x \right\}$$ is fractional part so it is lies between 0 to 1($$0 \le \left\{ x \right\} < 1$$).

By considering positive sign, we get

$$0 \le {{ - 2 + \sqrt {4 + 12{a^2}} } \over { - 6}} < 1$$

$$ \Rightarrow $$$$0 \ge - 2 + \sqrt {4 + 12{a^2}} > - 6$$

$$ \Rightarrow $$$$2 \ge + \sqrt {4 + 12{a^2}} > - 4$$

$$\because$$$$ + \sqrt {4 + 12{a^2}} $$ is always positive which is greater than any negative no. So can ignore the inequality $$ + \sqrt {4 + 12{a^2}} > - 4$$

Consider this inequality,
$$2 \ge + \sqrt {4 + 12{a^2}} $$
$$ \Rightarrow $$ $$4 \ge 4 + 12{a^2}$$
$$ \Rightarrow $$ $$12{a^2} \le 0$$
$$ \Rightarrow $$ $${a^2} \le 0$$
$$ \Rightarrow $$ $${a^2} = 0$$
$$ \Rightarrow $$ $${a} = 0$$

If $$a$$ = 0 then $$ - 3{\left\{ x \right\}^2} + 2\left\{ x \right\} = 0$$ so $$\left\{ x \right\}$$ becomes 0 but question says $$\left\{ x \right\}$$ $$ \ne $$ 0.
So $$a$$ can't be 0.

Now by considering negative sign, we get


$$0 \le {{ - 2 - \sqrt {4 + 12{a^2}} } \over { - 6}} < 1$$

$$ \Rightarrow $$$$0 \ge - 2 - \sqrt {4 + 12{a^2}} > - 6$$

$$ \Rightarrow $$$$2 \ge - \sqrt {4 + 12{a^2}} > - 4$$

As 2 is always greater than $${ - \sqrt {4 + 12{a^2}} }$$. Ignore this inequality.

Now consider this inequality,

$$ - \sqrt {4 + 12{a^2}} > - 4$$

$$ \Rightarrow $$ $$\sqrt {4 + 12{a^2}} < 4$$

$$ \Rightarrow $$ $$4 + 12{a^2} < 16$$

$$ \Rightarrow $$ $$12{a^2} < 12$$

$$ \Rightarrow $$ $${a^2} < 1$$

$$ \Rightarrow $$ $$\left( {{a^2} - 1} \right) < 0$$

$$ \Rightarrow $$ $$\left( {a + 1} \right)\left( {a - 1} \right) < 0$$

$$ \Rightarrow $$ $$ - 1 < a < 1$$

But earlier we found that $$a$$ $$ \ne $$ 0.

So, the range of $$a$$ is = $$\left( { - 1,0} \right) \cup \left( {0,1} \right)$$

Given equations are

$$\,\,\,\,\,\,\,\,\,\,\,\,{x^2} + 2x + 3 = 0\,\,\,\,\,...\left( i \right)$$

$$\,\,\,\,\,\,\,\,\,\,\,\,a{x^2} + bx + c = 0\,\,\,...\left( {ii} \right)$$

Roots of equation $$(i)$$ are imaginary roots.

According to the question $$(ii)$$ will also have both roots same as $$(i).$$

Thus $${a \over 1} = {b \over 2} = {c \over 3} = \lambda \left( {say} \right)$$

$$ \Rightarrow a = \lambda ,b = 2\lambda ,c = 3\lambda $$

Hence, required ratio is $$1:2:3$$