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Grade: 11
        
how to find the locus of the midpoints of the chords of an ellipse of specific length
11 months ago

Answers : (1)

Arun
15912 Points
							
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)
11 months ago
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