The locus of the point of inter section of perpendicular chords passing through the foci of the ellipse `x^2/a^2+y^2/b^2=1(a >b)` is

Options

A. `x ^(2) +y ^(2) =a ^(2)`
B. `x ^(2) +y ^(2) =a ^(2) -b ^(2)`
C. `x ^(2) + y ^(2) =b ^(2)`
D. `x ^(2) +y ^(2) =a ^(2) +b ^(2)`

Answer

To find the locus of the point of intersection of perpendicular chords passing through the foci of the ellipse given by the equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) (where \( a > b \)), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Foci of the Ellipse:**
The foci of the ellipse are located at the points \( (ae, 0) \) and \( (-ae, 0) \), where \( e \) is the eccentricity of the ellipse. The eccentricity \( e \) is given by:
\[ e = \sqrt{1 - \frac{b^2}{a^2}} \]

2. **Let the Point of Intersection be \( (h, k) \):**
We denote the point of intersection of the chords as \( (h, k) \).

3. **Determine the Slopes of the Chords:**
Since the chords are perpendicular, we can denote one chord as passing through the focus \( (ae, 0) \) and the other through \( (-ae, 0) \). The slopes of these chords can be expressed as:
- For the chord through \( (ae, 0) \):
\[ \text{slope of chord} = \frac{k - 0}{h - ae} = \frac{k}{h - ae} \]
- For the chord through \( (-ae, 0) \):
\[ \text{slope of chord} = \frac{k - 0}{h + ae} = \frac{k}{h + ae} \]

4. **Set the Product of the Slopes to -1:**
Since the chords are perpendicular, we have:
\[ \frac{k}{h - ae} \cdot \frac{k}{h + ae} = -1 \]
Simplifying this gives:
\[ \frac{k^2}{(h - ae)(h + ae)} = -1 \]
Thus,
\[ k^2 = - (h^2 - a^2e^2) = a^2e^2 - h^2 \]

5. **Substitute the Value of \( e^2 \):**
From the definition of eccentricity, we have:
\[ e^2 = 1 - \frac{b^2}{a^2} \]
Therefore,
\[ a^2e^2 = a^2(1 - \frac{b^2}{a^2}) = a^2 - b^2 \]

6. **Combine the Results:**
Substituting \( a^2e^2 \) into the equation for \( k^2 \):
\[ k^2 = (a^2 - b^2) - h^2 \]
Rearranging gives:
\[ h^2 + k^2 = a^2 - b^2 \]

7. **Replace \( h \) and \( k \) with \( x \) and \( y \):**
Finally, we replace \( h \) with \( x \) and \( k \) with \( y \):
\[ x^2 + y^2 = a^2 - b^2 \]

### Conclusion:
The locus of the point of intersection of the perpendicular chords passing through the foci of the ellipse is given by:
\[ x^2 + y^2 = a^2 - b^2 \]