Multiple choice

The locus of the middle points of the system of chords of the circle $x^2+y^2=16$ which are parallel to the line $2y=4x+5$ is ?

  1. $x=2y$
  2. $x+2y=0$
  3. $y+2x=0$
  4. $y=2x$
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B Correct answer
Explanation

The line 2y = 4x + 5 has slope 2. Chords parallel to this line have slope 2. The locus of midpoints of chords with slope m for a circle x^2 + y^2 = r^2 is y = -x/m. Here m = 2, so y = -x/2, or x + 2y = 0.

AI explanation

The given circle is centered at the origin (0,0) with a radius of 4. The slope of the given line 2y = 4x + 5 is 2, meaning the parallel chords also have a slope of 2. The locus of the midpoints of parallel chords is a diameter that is perpendicular to the chords and passes through the center. A line perpendicular to the chords will have a slope of -0.5, and passing through (0,0), its equation is y = -0.5x, which rearranges to x + 2y = 0.

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