Point C: y=0 ⟹ ax=(a2−b2)cosuy=0\implies ax=(a^2-b^2)cosuy=0⟹ax=(a2−b2)cosu
Point D: x=0 ⟹ by=(a2−b2)sinux=0\implies by=(a^2-b^2)sinux=0⟹by=(a2−b2)sinu
Midpoint of CD: xM=(a2−b2)cosu2a,yM=(a2−b2)sinu2bx_M=\frac{(a^2-b^2)cosu}{2a}, y_M=\frac{(a^2-b^2)sinu}{2b}xM=2a(a2−b2)cosu,yM=2b(a2−b2)sinu
cos2u+sin2u=1cos^2u+sin^2u=1cos2u+sin2u=1
4a2xM2(a2−b2)2+4b2yM2(a2−b2)2=1\frac{4a^2x_M^2}{(a^2-b^2)^2}+\frac{4b^2y_M^2}{(a^2-b^2)^2}=1(a2−b2)24a2xM2+(a2−b2)24b2yM2=1
4a2xM2+4b2yM2=(a2−b2)24a^2x_M^2+4b^2y_M^2=(a^2-b^2)^24a2xM2+4b2yM2=(a2−b2)2
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