Repunits!!

I solved this way:

Fist, let rewrite the second member as follows:

$=2\times \underbrace { 111...111 }_{ y\quad times } +9\times ({ \underbrace { 111...111 }_{ y\quad times } })^{ 2 }$

Extracting the common factor and multiplying again we have:

$=\underbrace { 111...111 }_{ y\quad times } \times ({ 2+\underbrace {999...999 }_{ y\quad times } })$

But we have that ${ \underbrace { 999...999 }_{ y\quad times } }={ 10 }^{ y }-1$

Therefore, the second member becomes:

$=\underbrace { 111...111 }_{ y\quad times } \times ({ { { 10 }^{ y }+1 } })$

Applying the distributive propriety, we have that

$=\underbrace { 111...111 }_{ y\quad times } \times { 10 }^{ y }+\underbrace { 111...111 }_{ y\quad times }$

$=\underbrace { \underbrace { 111...111 }_{ y\quad times } \underbrace { 000...000 }_{ y\quad times } }_{ 2y } +\underbrace { 111...111 }_{ y\quad times }$

Wich is equal to:

$=\underbrace { 111...111 }_{ 2y }$

Hence,

$\underbrace { 111...111 }_{ x } =\underbrace { 111...111 }_{ 2y }$

Finally, we must have

$x=2\times{y}$ ,

Therefore, $\frac { x }{ y } =\boxed{2}$

$\quad \quad \quad Q.E.D.$

Starting from a smaller pattern:

$11=2+3^{2},$ where $x=2, y=1$

$1111=22+33^{2},$ where $x=4, y=2$

$111111=222+333^{2},$ where $x=6, y=3$

This will continue forever with the pattern of $y=\frac{1}{2}x$

So, the answer must be $\frac{x}{y}=\boxed{2}$

i madde use of G.P. 111....111(x times) = 1 + 10 + 100+ 1000 ...... and so on with finally x terms and common ratio 10 so 111....111(x times) = ((10 ^ x) - 1) / 9 thus L.H.S. = (( 10 ^ x ) - 1)/9 R.H.S. = 2*( ( 10 ^ y - 1) / 9 ) + 9 * { ( 10 ^ y - 1) / 9 } ^ 2 and solve

Simple approach is 2+3^2 is 11 so x=2 and y=1. Therefore x/y=2.