Mathematics
Mathematics
Problem
From the points on the circle , tangents are drawn to the hyperbola ; prove that the locus of the middle-points of the chords of contact is the curve .Intuitive explanation
The key is understanding that a chord of contact is the line joining the two tangent points on the hyperbola when tangents are drawn from an external point. For any point on the circle, you'll find two tangent lines to the hyperbola; the line connecting those tangent points is the chord of contact. Rather than finding tangent points directly, use the chord-of-contact equation and the constraint that the external point lies on the circle to derive a relationship for the midpoint. The locus emerges by eliminating the external point's coordinates and keeping only the midpoint's.
Intuitive explanation
The key is understanding that a chord of contact is the line joining the two tangent points on the hyperbola when tangents are drawn from an external point. For any point on the circle, you'll find two tangent lines to the hyperbola; the line connecting those tangent points is the chord of contact. Rather than finding tangent points directly, use the chord-of-contact equation and the constraint that the external point lies on the circle to derive a relationship for the midpoint. The locus emerges by eliminating the external point's coordinates and keeping only the midpoint's.