If (1,–2) is the focus and $+\mathrm{y}-2=0$ is the directix of the  ellipse $17{\mathrm{x}}^2-2\mathrm{xy}+17{\mathrm{y}}^2-32\mathrm{x}+76\mathrm{y}+86=0$ then its  eccentricity is

Given focus =(1, -2) and equation of directrix is $\mathrm{x}+\mathrm{y}-2=0$ 

Now let P(x, y) be any point on ellipse then equation of ellipse is 

${(\mathrm{x}-1)}^2+{(\mathrm{y}+2)}^2=\frac{{\mathrm{e}}^2{(\mathrm{x}+\mathrm{y}-2)}^2}{2}$

$\Rightarrow 2\left[{\mathrm{x}}^2+1-2\mathrm{x}+{\mathrm{y}}^2+4+4\mathrm{y}\right]={\mathrm{e}}^2\left({\mathrm{x}}^2+{\mathrm{y}}^2+4+2\mathrm{xy}-4\mathrm{y}-4\mathrm{x}\right)$

$\Rightarrow {\mathrm{x}}^2(2-{\mathrm{e}}^2)-2{\mathrm{e}}^2\mathrm{xy}+(4{\mathrm{e}}^2-4)\mathrm{x}+(2-{\mathrm{e}}^2){\mathrm{y}}^2+(8+4{\mathrm{e}}^2)\mathrm{y}+10-4{\mathrm{e}}^2=0 -\left(1\right)$

Given equation of ellipse is 

$17{\mathrm{x}}^2-2\mathrm{xy}+17{\mathrm{y}}^2-32\mathrm{x}+76\mathrm{y}+86=0 -\left(2\right)$

 since (1)&(2) represents same line then

$\begin{array}{rcl}\frac{2-{\mathrm{e}}^2}{17}& =& \frac{2{\mathrm{e}}^2}{2}=\frac{4{\mathrm{e}}^2-4}{-32}\\ & \Rightarrow & 2-{\mathrm{e}}^2=17{\mathrm{e}}^2\\ & \Rightarrow & 18{\mathrm{e}}^2=2\\ & \Rightarrow & {\mathrm{e}}^2=\frac{1}{9}\\ & \Rightarrow & \boxed{\mathrm{e}=\frac{1}{3}}\end{array}$