Key Concepts and Formulas

• The line segment joining the center of a circle to the midpoint of a chord is perpendicular to the chord.
• The perpendicular bisector of a chord passes through the center of the circle.
• Distance formula: The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$.

Step-by-Step Solution

Step 1: Understand the problem and draw a diagram

We are given a circle with center at the origin and radius 3. We need to find the locus of the midpoints of chords that subtend an angle of $\frac{2\pi}{3}$ at the center. Let's visualize this with a diagram.

Step 2: Define the variables and points

• Let $O(0, 0)$ be the center of the circle.
• Let $r = 3$ be the radius of the circle. The equation of the circle is $x^2 + y^2 = 9$.
• Let $AB$ be a chord of the circle that subtends an angle $\angle AOB = \frac{2\pi}{3}$ at the center.
• Let $M(h, k)$ be the midpoint of the chord $AB$. We want to find the locus of $M$.

Step 3: Use the property that OM is perpendicular to AB

Since $M$ is the midpoint of the chord $AB$, the line segment $OM$ is perpendicular to $AB$. Also, $OM$ bisects the angle $\angle AOB$.

Step 4: Find the angle AOM

Since $OM$ bisects $\angle AOB$, we have: $\angle AOM = \frac{1}{2} \angle AOB = \frac{1}{2} \left(\frac{2\pi}{3}\right) = \frac{\pi}{3}$

Step 5: Consider the right-angled triangle OMA and use trigonometry

In the right-angled triangle $\triangle OMA$, we have $\angle OMA = 90^\circ$, $OA = r = 3$, and $\angle AOM = \frac{\pi}{3}$. We want to find the length of $OM$. Using cosine: $\cos(\angle AOM) = \frac{OM}{OA}$ $\cos\left(\frac{\pi}{3}\right) = \frac{OM}{3}$ Since $\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}$, we have: $\frac{1}{2} = \frac{OM}{3}$ $OM = \frac{3}{2}$

Step 6: Find the locus of the midpoint M(h, k)

The distance $OM$ is constant and equal to $\frac{3}{2}$. Using the distance formula between $O(0, 0)$ and $M(h, k)$: $OM = \sqrt{(h-0)^2 + (k-0)^2} = \sqrt{h^2 + k^2}$ Since $OM = \frac{3}{2}$: $\sqrt{h^2 + k^2} = \frac{3}{2}$ Squaring both sides: $h^2 + k^2 = \frac{9}{4}$

Step 7: Write the equation of the locus

Replacing $h$ with $x$ and $k$ with $y$, we get the equation of the locus: $x^2 + y^2 = \frac{9}{4}$

Common Mistakes & Tips

• Failing to recognize that the line from the center to the midpoint of the chord is perpendicular to the chord.
• Incorrectly calculating the angle $\angle AOM$.
• Making errors with trigonometric values.

Summary

We found the locus of the midpoints of the chords of the given circle that subtend an angle of $\frac{2\pi}{3}$ at the center. By using the geometric properties of chords and basic trigonometry, we found the locus to be another circle with equation $x^2 + y^2 = \frac{9}{4}$.

The final answer is \boxed{\frac{9}{4}}, which corresponds to option (D).