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The locus of mid-points of focal chords ...

The locus of mid-points of focal chords of the ellipse `(x^2)/(a^2)+(y^2)/(b^2)=1` is

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Let the midpoint of the focal chord the given ellipse be (h,k) . Then its equation is
`(hx)/(k)+(ky)/(b^(2))=(h^(2))/(a^(2))+(k^(2))/(b^(2))" " ["Using"T=S_(1)]`
Since this passes thoughb (ae,0), we have
or `(hae)/(a)=(h^(2))/(a^(2))+(k^(2))/(b^(2))`
Therefore, the locus of (h,k) is `(ex)/(a)=(x^(2))/(a^(2))+(y^(2))/(b^(2))`
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