Question

The locus of the mid-point of a chord of the circle \( x^2 + y^2 = 4 \), which subtends a right angle at the origin is

• A. \( x + y = 2 \)
• B. \( x^2 + y^2 = 1 \)
• C. \( x^2 + y^2 = 2 \)
• D. \( x + y = 1 \)

Hint:

For any circle \(x^2 + y^2 = r^2\), the locus of the midpoint of a chord subtending an angle \(\theta\) at the center is always a circle \(x^2 + y^2 = (r \cos \frac{\theta}{2})^2\).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
A chord of a circle subtending a right angle at the origin creates a right-angled triangle with the origin and the two endpoints of the chord. The distance from the origin to the midpoint of the chord can be found using trigonometry or the properties of right-angled triangles.

Step 2: Key Formula or Approach:
Let the midpoint of the chord be \(M(h, k)\). The distance of the midpoint from the origin is \(OM = \sqrt{h^2 + k^2}\). For a chord subtending \(90^\circ\) at the center, the distance \(d\) from the center to the chord is related to the radius \(r\) by: \[ d = r \cos(45^\circ) = \frac{r}{\sqrt{2}} \]

Step 3: Detailed Explanation:
Given the circle \(x^2 + y^2 = 4\), the radius \(r = 2\). 1. The chord subtends \(90^\circ\) at the origin. 2. The midpoint \(M(h, k)\) and the center \(O(0,0)\) form a line perpendicular to the chord. 3. In the right-angled triangle formed by the origin, the midpoint, and one endpoint of the chord, the angle at the origin is \(45^\circ\) (half of \(90^\circ\)). 4. Therefore, \(OM = r \cos(45^\circ)\): \[ \sqrt{h^2 + k^2} = 2 \cdot \frac{1}{\sqrt{2}} = \sqrt{2} \] 5. Squaring both sides: \[ h^2 + k^2 = 2 \] 6. Replacing \((h, k)\) with \((x, y)\), we get the locus: \[ x^2 + y^2 = 2 \]

Step 4: Final Answer
The locus of the midpoint is \( x^2 + y^2 = 2 \).