Multiple choice

Let $C$ be the circle with centre $\left( {0,0} \right)$ and radius $3$ units. The equation of the locus of the mid-points of the chords of the circle $C$ that subtend an angles of $\frac{{2\pi }}{3}$ at its centre, is:

  1. ${x^3} + {y^2} = 1$
  2. ${x^2} + {y^2} = \frac{{27}}{4}$
  3. ${x^2} + {y^2} = \frac{9}{4}$
  4. ${x^2} + {y^2} = \frac{3}{2}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

For a chord of a circle with radius R subtending angle 2*theta at the center, the distance from the center to the chord is d = R * cos(theta). Here R = 3 and 2*theta = 2*pi/3, so theta = pi/3. d = 3 * cos(pi/3) = 3 * 0.5 = 1.5. The locus of the midpoints is a circle with radius d = 1.5, so x^2 + y^2 = (1.5)^2 = 2.25 = 9/4.

AI explanation

Let the midpoint of the chord be M and the center of the circle be O. In triangle OMC, the angle subtended at the center is 2pi by 3, and OM is perpendicular to the chord, creating a right triangle with angle pi by 3 at the center. Using trigonometry, cos(pi by 3) equals OM divided by the radius 3, so OM equals 3 divided by 2. The locus of M is a circle with radius 3 divided by 2, so its equation is x squared plus y squared equals 9 by 4.

Keep practicing — related questions