Multiple choice

The coordinates of the middle points of the chord cut-off by $2x - y + 18 = 0$ by the circle $x^2 + y^2 - 6x + 2y - 54 = 0$ are

  1. $(-7,4)$
  2. $(2,4)$
  3. $(4,1)$

  4. none of these
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A Correct answer
Explanation

The midpoint of a chord is the projection of the circle's center onto the line. Center is (3, -1). The line perpendicular to 2x - y + 18 = 0 passing through (3, -1) is x + 2y - 1 = 0. Solving the system 2x - y + 18 = 0 and x + 2y - 1 = 0 gives (-7, 4).

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