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Show that the midpoints of focal chords ...

Show that the midpoints of focal chords of a hyperbola `(x^2)/(a^2)-(y^2)/(b^2)=1` lie on another similar hyperbola.

Text Solution

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Let the chord AB be bisected at point `P(h,k).`
So, equation of chord AB is
`(hx)/(a^(2))-(ky)/(b^(2))=(h^(2))/(a^(2))-(k^(2))/(b^(2))" (Using T = S"_(1)")"`
Let it pass through the focus (ae, 0).
(1) `therefore" "(he)/(a)=(h^(2))/(a^(2))-(k^(2))/(b^(2))`
Therefore, locus of point P is
(2) `(x^(2))/(a^(2))-(y^(2))/(b^(2))=(ex)/(a)`
`rArr" "(1)/(a^(2))[x^(2)-aex]-(y^(2))/(b^(2))=0`
`rArr" "((x-(ae)/(2))^(2))/(a^(2))-(y^(2))/(b^(2))=(e^(2))/(4)`
This is also hyperbola with eccentricity e.
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