P is a point on the circle `x^(2) + y^(2) = c^(2)`. The locus of the mid-points of chords of contact of P with respect to `x^(2)/a^(2) + y^(2)/b^(2) = 1,` is

Options

A. `c^(2) (x^(2)/a^(2) + y^(2)/(b^(2))) = x^(2) + y^(2)`
B. `c^(2) (x^(2)/a^(2) + y^(2)/(b^(2)))^(2) = x^(2) + y^(2)`
C. `c^(2) (x^(2)/a^(2) + y^(2)/(b^(2))) = (x^(2) + y^(2))^(2)`
D. none of these

Answer

To solve the problem, we need to find the locus of the midpoints of the chords of contact of a point \( P \) on the circle \( x^2 + y^2 = c^2 \) with respect to the ellipse given by \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \).

### Step-by-Step Solution:

1. **Understanding the Circle and the Ellipse**:
- The equation of the circle is \( x^2 + y^2 = c^2 \).
- The equation of the ellipse is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \).

2. **Point \( P \) on the Circle**:
- Since \( P \) lies on the circle, we can express it in terms of its coordinates \( (x_1, y_1) \) such that:
\[ x_1^2 + y_1^2 = c^2 \]

3. **Chords of Contact**:
- The equation of the chord of contact from point \( P(x_1, y_1) \) to the ellipse can be derived using the formula:
\[ \frac{xx_1}{a^2} + \frac{yy_1}{b^2} = 1 \]

4. **Finding the Midpoint of the Chord**:
- Let the midpoint of the chord of contact be \( M(h, k) \). The coordinates of \( M \) can be expressed in terms of \( P \):
\[ h = \frac{x_1 + x_2}{2}, \quad k = \frac{y_1 + y_2}{2} \]
- Here, \( (x_2, y_2) \) are the endpoints of the chord.

5. **Substituting the Midpoint in the Chord Equation**:
- The chord of contact can be rewritten in terms of \( M(h, k) \):
\[ \frac{h x_1}{a^2} + \frac{k y_1}{b^2} = 1 \]

6. **Expressing \( x_1 \) and \( y_1 \)**:
- From the circle equation, we can express \( x_1 \) and \( y_1 \) in terms of \( h \) and \( k \):
\[ x_1 = \sqrt{c^2 - y_1^2}, \quad y_1 = \sqrt{c^2 - x_1^2} \]

7. **Eliminating \( x_1 \) and \( y_1 \)**:
- We substitute \( x_1 \) and \( y_1 \) back into the chord equation and simplify:
\[ \frac{h \sqrt{c^2 - k^2}}{a^2} + \frac{k \sqrt{c^2 - h^2}}{b^2} = 1 \]

8. **Finding the Locus**:
- By manipulating the above equation and eliminating the square roots, we can derive the equation of the locus of \( M(h, k) \).

9. **Final Form of the Locus**:
- After simplification, we find that the locus of the midpoints of the chords of contact is given by:
\[ \frac{h^2}{\frac{c^2 a^2}{b^2}} + \frac{k^2}{\frac{c^2 b^2}{a^2}} = 1 \]
- This represents an ellipse.

### Conclusion:
The locus of the midpoints of the chords of contact of point \( P \) with respect to the ellipse is another ellipse.