Multiple choice

The locus of the mid points of the chords of the circle $x^{2} + y^{2} + 4x - 6y - 12 = 0$ which subtend an angle of $\pi/3$ radians at its circumference is

  1. $(x + 2)^{2} + (y - 3)^{2} = 6.25$
  2. $(x - 2)^{2} + (y + 3)^{2} = 6.25$
  3. $(x + 2)^{2} + (y - 3)^{2} = 18.75$
  4. $(x + 2)^{2} + (y + 3)^{2} = 18.75$
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A Correct answer
Explanation

The circle is (x+2)^2 + (y-3)^2 = 25. Radius R=5. A chord subtending pi/3 at circumference subtends 2*pi/3 at center. The distance d from center to chord is R*cos(pi/3) = 5 * 0.5 = 2.5. The locus of midpoints of chords at distance d from center is a circle with radius d, centered at the same point: (x+2)^2 + (y-3)^2 = (2.5)^2 = 6.25.

AI explanation

The circle equation is x^2 + y^2 + 4x - 6y - 12 = 0, with center at (-2, 3) and radius R = 5. A chord subtending an angle of 60 degrees at the circumference subtends a central angle of 120 degrees. Using the right triangle formed by the radius, half-chord, and perpendicular distance d from the center, we get d = R times the sine of 30 degrees, which equals 2.5. Squaring this distance gives 6.25, so the locus is a concentric circle with equation (x + 2)^2 + (y - 3)^2 = 6.25.

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