Let C be the circle with centre (0, 0) and radius 3 units. The equation of the locus of the mid points of the chords of the circle C that subtend an angle of `(2pi)/3` at its center is

Options

A. `x^(2)+y^(2)=(3)/(2)`
B. `x^(2)+y^(2)=1`
C. `x^(2)+y^(2)=(27)/(4)`
D. `x^(2)+y^(2)=(9)/(4)`

Answer

To find the equation of the locus of the midpoints of the chords of the circle \( C \) that subtend an angle of \( \frac{2\pi}{3} \) at its center, we can follow these steps:

### Step 1: Understand the Circle
The circle \( C \) has its center at \( (0, 0) \) and a radius of \( 3 \) units. The equation of the circle is given by:
\[ x^2 + y^2 = 3^2 = 9 \]

### Step 2: Identify the Angle Subtended
The angle subtended by the chords at the center is \( \frac{2\pi}{3} \) radians, which is equivalent to \( 120^\circ \).

### Step 3: Determine the Midpoint Distance
The midpoint of the chord that subtends an angle of \( 120^\circ \) at the center will be at a fixed distance from the center. To find this distance, we can use the properties of triangles.

In a triangle formed by the radius and the chord, we can consider half of the subtended angle:
\[ \text{Half of } 120^\circ = 60^\circ \]
The radius of the circle is \( 3 \) units. The distance from the center to the midpoint of the chord can be found using the sine function in a \( 30-60-90 \) triangle:
\[ \text{Distance from center to midpoint} = r \cdot \cos\left(\frac{120^\circ}{2}\right) = 3 \cdot \cos(60^\circ) = 3 \cdot \frac{1}{2} = \frac{3}{2} \]

### Step 4: Write the Equation of the Locus
The locus of the midpoints of the chords will form a circle with the center at \( (0, 0) \) and a radius equal to \( \frac{3}{2} \). The equation of this circle is:
\[ x^2 + y^2 = \left(\frac{3}{2}\right)^2 = \frac{9}{4} \]

### Final Answer
Thus, the equation of the locus of the midpoints of the chords that subtend an angle of \( \frac{2\pi}{3} \) at the center is:
\[ x^2 + y^2 = \frac{9}{4} \]