If the ellipse `x^2/a^2+y^2/b^2=1` and the circle `x^2+y^2=r^2` where `bltrlta` intersect in four points and the slope of a common tangent to the ellipse and the circle is `b/a` , then `r^2` is equal to

Options

A. `(2b^2)/(a^2+b^2)`
B. `(2a^2b^2)/(a^2+b^2)`
C. `(2a^2)/(a^2+b^2)`
D. `(2a^2b^2)/(a^2-b^2)`

Answer

To solve the problem, we need to find the value of \( r^2 \) given the equations of the ellipse and the circle, along with the slope of the common tangent.

### Step-by-Step Solution:

1. **Identify the Equations**:
- The equation of the ellipse is given by:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
- The equation of the circle is:
\[ x^2 + y^2 = r^2 \]

2. **Common Tangent Slope**:
- The slope of the common tangent to both the ellipse and the circle is given as \( \frac{b}{a} \).

3. **Equation of Tangent to the Ellipse**:
- The equation of the tangent to the ellipse at a slope \( m \) is:
\[ y = mx \pm \sqrt{a^2m^2 + b^2} \]
- Substituting \( m = \frac{b}{a} \):
\[ y = \frac{b}{a}x \pm \sqrt{a^2\left(\frac{b}{a}\right)^2 + b^2} \]
- Simplifying the square root:
\[ \sqrt{b^2 + b^2} = \sqrt{2b^2} = b\sqrt{2} \]
- Thus, the tangent line becomes:
\[ y = \frac{b}{a}x \pm b\sqrt{2} \]

4. **Equation of Tangent to the Circle**:
- The equation of the tangent to the circle at slope \( m \) is:
\[ y = mx \pm \sqrt{r^2(1 + m^2)} \]
- Substituting \( m = \frac{b}{a} \):
\[ y = \frac{b}{a}x \pm \sqrt{r^2\left(1 + \left(\frac{b}{a}\right)^2\right)} \]
- This simplifies to:
\[ y = \frac{b}{a}x \pm \sqrt{r^2\left(\frac{a^2 + b^2}{a^2}\right)} = \frac{b}{a}x \pm \frac{r\sqrt{a^2 + b^2}}{a} \]

5. **Setting the Tangents Equal**:
- For the tangents to be the same, the constant terms must be equal:
\[ b\sqrt{2} = \pm \frac{r\sqrt{a^2 + b^2}}{a} \]
- Taking the positive case (as both tangents should be equal):
\[ b\sqrt{2} = \frac{r\sqrt{a^2 + b^2}}{a} \]

6. **Solving for \( r \)**:
- Rearranging gives:
\[ r = \frac{ab\sqrt{2}}{\sqrt{a^2 + b^2}} \]

7. **Finding \( r^2 \)**:
- Squaring both sides:
\[ r^2 = \frac{2a^2b^2}{a^2 + b^2} \]

### Conclusion:
Thus, the value of \( r^2 \) is:
\[ r^2 = \frac{2a^2b^2}{a^2 + b^2} \]