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how to find the locus of the midpoints of the chords of an ellipse of specific length

how to find the locus of the midpoints of the chords of an ellipse of specific length

Grade:11

1 Answers

Arun
25763 Points
3 years ago
Dear Amandeep
 

Let M(X,Y) be the mid-point of a chord whose distance from O is c.

The diameter OM has gradient Y/X

The conjugate diameter has gradient m where m(Y/X) = −b²/a² → m = −b²X/(a²Y)

All chords || to conjugate diameter are bisected by OM. Hence chord has slope m

∴ equation of chord is y−Y = {−b²X/(a²Y)}(x−X) → xX/a² + yY/b² − X²/a² − Y²/b² = 0

If chord is at distance c from O then c = |X²/a²+Y²/b²| / √X²/a⁴+Y²/b⁴)

Squaring gives (X²/a²+Y²/b²)² = c²(X²/a⁴+Y²/b⁴) for equation of locus 
 

Regards
Arun (askIITians forum expert)

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