Which Midpoints Are Possible?

★ 6/10Midpoint Chord Method conic sections analytic geometry Casework Discriminant

Problem

Let $A, B$ be two points on the hyperbola $x^2 - \dfrac{y^2}{9} = 1$ . For each of the points

(1, 1), \qquad (-1, 2), \qquad (1, 3), \qquad (-1, -4),

determine whether it can be the midpoint of segment $AB$ , with proof.

Answer

Solution

Difficulty	6/10
Topics	Midpoint Chord Method, conic sections, analytic geometry, Casework, Discriminant

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