Let $C _ { 1 }$ be the circle of radius 1 with center at the origin. Let $C _ { 2 }$ be the circle of radius $r$ with center at the point $A = ( 4,1 )$, where $1 < r < 3$. Two distinct common tangents $P Q$ and $S T$ of $C _ { 1 }$ and $C _ { 2 }$ are drawn. The tangent $P Q$ touches $C _ { 1 }$ at $P$ and $C _ { 2 }$ at $Q$. The tangent $S T$ touches $C _ { 1 }$ at $S$ and $C _ { 2 }$ at $T$. Mid points of the line segments $P Q$ and $S T$ are joined to form a line which meets the $x$-axis at a point $B$. If $A B = \sqrt { 5 }$, then the value of $r ^ { 2 }$ is
Let $C _ { 1 }$ be the circle of radius 1 with center at the origin. Let $C _ { 2 }$ be the circle of radius $r$ with center at the point $A = ( 4,1 )$, where $1 < r < 3$. Two distinct common tangents $P Q$ and $S T$ of $C _ { 1 }$ and $C _ { 2 }$ are drawn. The tangent $P Q$ touches $C _ { 1 }$ at $P$ and $C _ { 2 }$ at $Q$. The tangent $S T$ touches $C _ { 1 }$ at $S$ and $C _ { 2 }$ at $T$. Mid points of the line segments $P Q$ and $S T$ are joined to form a line which meets the $x$-axis at a point $B$. If $A B = \sqrt { 5 }$, then the value of $r ^ { 2 }$ is