SAT1000 - P845

Geometry Level 4

As shown above, the parabola $E: y^2=x$ and circle $M: (x-4)^2+y^2=r^2$ for $r>0$ intersect at points $A$ , $B$ , $C$ , and $D$ ; and their relative positions are shown in the figure. Lines $AC$ and $BD$ intersect at point $P$ .

Find the coordinates $(x_0,0)$ of $P$ as the area of quadrilateral $ABCD$ reaches the maximum when $r$ varies. Submit $\lfloor 1000x_0 \rfloor$ .

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