Grade 11Analytical Geometry
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
1 Answer
Arun
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
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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