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Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) §3.5 Parabolas, Ellipses, and Hyperbolas, #30 · Geometry · Conic sections

Figure B proves the reflecting property of an ellipse

This problem is about parabolas, are about ellipses, are about hyperbolas.

Figure B proves the reflecting property of an ellipse. $R$ is the mirror image of $F_{1}$ in the tangent line; $Q$ is any other point on the line. Deduce steps 2,3,4 from 1,2,3:
1. $P F_{1} + P F_{2} < Q F_{1} + Q F_{2}$ (left side $= 2 a$ , $Q$ is outside)
2. $P R + P F_{2} < QR + Q F_{2}$
3. $P$ is on the straight line from $F_{2}$ to $R$
4. $α = β$ : the reflecting property is proved.

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Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) · §3.5 Parabolas, Ellipses, and Hyperbolas, #30

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