Find the locus of the mid-point of the chord of the circle x2  + y2  = a2 which subtends a 90 degree angle at point (p,q) lying inside the circle . 


To find the locus of the midpoint of the chord of the circle \(x^2 + y^2 = a^2\), which subtends a 90° angle at the point \((p, q)\) inside the circle, let's go through the steps systematically.

### 1. **Equation of the Circle**
The given equation of the circle is:
\[
x^2 + y^2 = a^2
\]
This represents a circle with center at the origin \((0, 0)\) and radius \(a\).

### 2. **Chord Subtending 90° at Point (p, q)**
Let the chord \(AB\) subtend a 90° angle at the point \((p, q)\), which lies inside the circle. The chord’s midpoint will be denoted as \(M(x_1, y_1)\).

### 3. **Perpendicularity Condition**
For a chord to subtend a 90° angle at a point, the line joining the point \((p, q)\) to the midpoint \(M(x_1, y_1)\) of the chord must be perpendicular to the chord. Hence, the vector from \((p, q)\) to \(M(x_1, y_1)\) will be perpendicular to the vector representing the chord.

In other words, the slope of the line joining the point \((p, q)\) to \(M(x_1, y_1)\) and the slope of the chord are negative reciprocals.

### 4. **Power of the Point**
The power of a point \((p, q)\) with respect to a circle is given by:
\[
p^2 + q^2 - a^2
\]
This is also equal to the square of the length of the perpendicular drawn from the point \((p, q)\) to the chord.

If \(M(x_1, y_1)\) is the midpoint of the chord, the distance from the point \((p, q)\) to the midpoint \(M(x_1, y_1)\) can be related to the radius of the circle and the distance of \((p, q)\) from the center.

### 5. **Locus Equation**
Using the condition of perpendicularity and the power of a point, we can derive the equation of the locus of the midpoint \(M(x_1, y_1)\).

The required locus is the **circle** with the equation:
\[
(x - p)^2 + (y - q)^2 = \frac{a^2 - (p^2 + q^2)}{2}
\]
This is the equation of the locus of the midpoint of the chord that subtends a 90° angle at the point \((p, q)\).

### Summary
The locus of the midpoint of the chord that subtends a 90° angle at the point \((p, q)\) inside the circle \(x^2 + y^2 = a^2\) is a circle given by:
\[
(x - p)^2 + (y - q)^2 = \frac{a^2 - (p^2 + q^2)}{2}
\]

Would you like more details on how this result was derived?

### Relative Questions:
1. What is the significance of a chord subtending a specific angle at a point?
2. How can we generalize this result for angles other than 90°?
3. How does the power of a point affect the properties of chords in a circle?
4. Can this method be applied to ellipses or other conic sections?
5. What are some practical applications of locus problems in geometry?

### Tip:
When working with circle geometry, always try to use the power of a point and properties of perpendicularity to simplify the problem.

Find the locus of the mid-point of the chord of the circle x^2 + y^2 = a^2 which subtends a 90 degree angle at point (p,q) lying inside the circle.

Explore how to find the locus of the midpoint of a chord in a circle x^2 + y^2 = a^2, which subtends a 90° angle at a point (p, q) inside the circle. This problem involves key geometric concepts such as the perpendicularity condition and the power of a point. Suitable for students in Grades 10-12.

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