Let there are exactly two points on the ellipse `(x^(2))/(a^(2))+(y^(2))/(b^(2))=1` whose distance from (0, 0) are equal to `sqrt((a^(2))/(2)+b^(2))`. Then, the eccentricity of the ellipse is equal to

Options

A. `(1)/(2)`
B. `(1)/(2sqrt2)`
C. `(1)/(sqrt2)`
D. None of these

Answer

To solve the problem, we need to find the eccentricity of the ellipse given that there are exactly two points on the ellipse whose distance from the origin (0, 0) is equal to \(\sqrt{\frac{a^2}{2} + b^2}\).

### Step-by-Step Solution:

1. **Understanding the Ellipse Equation**:
The equation of the ellipse is given by:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
Here, \(a\) is the semi-major axis and \(b\) is the semi-minor axis.

2. **Distance from the Origin**:
We are given that the distance from the origin to the points on the ellipse is:
\[ d = \sqrt{\frac{a^2}{2} + b^2} \]

3. **Setting Up the Distance Equation**:
The distance from the origin to any point \((x, y)\) on the ellipse is given by:
\[ \sqrt{x^2 + y^2} \]
Therefore, we need to find points \((x, y)\) on the ellipse such that:
\[ \sqrt{x^2 + y^2} = \sqrt{\frac{a^2}{2} + b^2} \]

4. **Squaring Both Sides**:
Squaring both sides gives:
\[ x^2 + y^2 = \frac{a^2}{2} + b^2 \]

5. **Substituting for \(y^2\)**:
From the ellipse equation, we can express \(y^2\) in terms of \(x^2\):
\[ y^2 = b^2 \left(1 - \frac{x^2}{a^2}\right) \]
Substituting this into the distance equation:
\[ x^2 + b^2 \left(1 - \frac{x^2}{a^2}\right) = \frac{a^2}{2} + b^2 \]

6. **Simplifying the Equation**:
This simplifies to:
\[ x^2 + b^2 - \frac{b^2 x^2}{a^2} = \frac{a^2}{2} + b^2 \]
\[ x^2 \left(1 - \frac{b^2}{a^2}\right) = \frac{a^2}{2} \]

7. **Finding Conditions for Two Points**:
For there to be exactly two points, the coefficient of \(x^2\) must be positive:
\[ 1 - \frac{b^2}{a^2} > 0 \implies b^2 < a^2 \]

8. **Relating \(b^2\) and \(a^2\)**:
From the equation \(x^2 \left(1 - \frac{b^2}{a^2}\right) = \frac{a^2}{2}\), we can express \(b^2\):
\[ b^2 = a^2 - \frac{a^2}{2} = \frac{a^2}{2} \]

9. **Finding the Eccentricity**:
The eccentricity \(e\) of the ellipse is given by:
\[ e = \sqrt{1 - \frac{b^2}{a^2}} \]
Substituting \(b^2 = \frac{a^2}{2}\):
\[ e = \sqrt{1 - \frac{\frac{a^2}{2}}{a^2}} = \sqrt{1 - \frac{1}{2}} = \sqrt{\frac{1}{2}} = \frac{1}{\sqrt{2}} \]

### Final Answer:
The eccentricity of the ellipse is:
\[ e = \frac{1}{\sqrt{2}} \]