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Problem 1-2 (Geometry) [Difficulty 3] [PDF]
QUESTION
(Original) Define points $A\left(2,5+\sqrt{21}\right)$ , $B\left(2,5-\sqrt{21}\right)$ . Now let a point $P$ be initially at $P_{0}\left(4,5\right)$ . Let $Q$ be the point on ray $\overrightarrow{AP}$ such that $PB=PQ$ . As $P$ moves along the lower half of the ellipse $25x^{2}-100x+4y^{2}-40y+100=0$ (to eventually stop at $\left(0,5\right)$), the point $Q$ traces a path. Find the length of this path.
(Original) Define points $A\left(2,5+\sqrt{21}\right)$ , $B\left(2,5-\sqrt{21}\right)$ . Now let a point $P$ be initially at $P_{0}\left(4,5\right)$ . Let $Q$ be the point on ray $\overrightarrow{AP}$ such that $PB=PQ$ . As $P$ moves along the lower half of the ellipse $25x^{2}-100x+4y^{2}-40y+100=0$ (to eventually stop at $\left(0,5\right)$), the point $Q$ traces a path. Find the length of this path.
Overview/Laconic Solution Sketch
Apply geometric ellipse definition, and from that calculate arc length.
