20. As shown in the figure, the ellipse $\mathrm { E } : \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ $(a > b > 0)$ has eccentricity $\frac { \sqrt { 2 } } { 2 }$. A moving line $l$ passing through point $\mathrm { P } ( 0, 1 )$ intersects the ellipse at points $\mathrm { A }$ and $\mathrm { B }$. When line $l$ is parallel to the $x$-axis, the length of the chord intercepted by line $l$ on ellipse $E$ is $2 \sqrt { 2 }$. (1) Find the equation of ellipse $E$; (2) In the rectangular coordinate system $xOy$, does there exist a fixed point $Q$ different from point $P$ such that $\frac { | Q A | } { | Q B | } = \frac { | P A | } { | P B | }$ always holds? If it exists, find the coordinates of point $Q$; if it does not exist, explain the reason. [Figure]
20. As shown in the figure, the ellipse $\mathrm { E } : \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ $(a > b > 0)$ has eccentricity $\frac { \sqrt { 2 } } { 2 }$. A moving line $l$ passing through point $\mathrm { P } ( 0, 1 )$ intersects the ellipse at points $\mathrm { A }$ and $\mathrm { B }$. When line $l$ is parallel to the $x$-axis, the length of the chord intercepted by line $l$ on ellipse $E$ is $2 \sqrt { 2 }$.\\
(1) Find the equation of ellipse $E$;\\
(2) In the rectangular coordinate system $xOy$, does there exist a fixed point $Q$ different from point $P$ such that $\frac { | Q A | } { | Q B | } = \frac { | P A | } { | P B | }$ always holds? If it exists, find the coordinates of point $Q$; if it does not exist, explain the reason.\\
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