Multiple choice

The locus of the centre of the circles such that the point $(2, 3)$ is the midpoint of the chord $5x+2y=16$ is

  1. $2x-5y+11=0$
  2. $2x-5y-11=0$
  3. $2x+5y+11=0$
  4. $None$
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A Correct answer
Explanation

The chord is 5x + 2y = 16. The center (h, k) of the circle must lie on a line perpendicular to the chord passing through the midpoint (2, 3). The slope of the chord is -5/2, so the perpendicular slope is 2/5. Equation: y - 3 = (2/5)(x - 2) => 5y - 15 = 2x - 4 => 2x - 5y + 11 = 0.

AI explanation

The locus of the centre of the desired circles is the line perpendicular to the given chord passing through the point (2, 3). The slope of the line 5x + 2y = 16 is -5/2, so the required perpendicular slope is 2/5. Using the point-slope form with point (2, 3), we write the equation as y - 3 = (2/5)(x - 2). Multiplying by 5 and rearranging gives 5y - 15 = 2x - 4, which simplifies to 2x - 5y + 11 = 0.

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