Multiple choice

The equation of diameter of the circle $\displaystyle (x - 2)^2 + (y + 1)^2 = 16$ which bisect the chord cut off by the circle on the line $\displaystyle x - 2y - 3 = 0$ is

  1. $\displaystyle 2x - y + 3 = 0$
  2. $\displaystyle 2x + y - 3 = 0$
  3. $\displaystyle 2x + y + 3 = 0$

  4. None of these
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B Correct answer
Explanation

The center of the circle is (2, -1). The diameter must pass through this center. The midpoint of the chord on the line x - 2y - 3 = 0 is found by dropping a perpendicular from the center (2, -1) to the line. The line perpendicular to x - 2y - 3 = 0 is 2x + y + k = 0. Since it passes through (2, -1), 2(2) + (-1) + k = 0, so k = -3. The diameter is 2x + y - 3 = 0.

AI explanation

The required diameter must pass through the center of the circle and be perpendicular to the given chord. The center of the circle (x - 2)^2 + (y + 1)^2 = 16 is (2, -1), and the slope of the chord x - 2y - 3 = 0 is 1/2. The slope of the perpendicular diameter is the negative reciprocal, which is -2. Using the point-slope form, the equation of the line passing through (2, -1) with slope -2 is y - (-1) = -2(x - 2), which simplifies to 2x + y - 3 = 0.

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