An ellipse of eccentricity `(2sqrt2)/3` is inscribed in a circle and a point within the circle is chosen at random. Let the probability that this point lies outside the ellipse be p. Then the value of 105 p is

Answer

To solve the problem, we need to find the probability \( p \) that a randomly chosen point within a circle lies outside an inscribed ellipse. We will then compute \( 105p \).

### Step-by-step Solution:

1. **Understanding the Problem**:
We have a circle with radius \( a \) and an ellipse inscribed in it. The eccentricity of the ellipse is given as \( e = \frac{2\sqrt{2}}{3} \).

2. **Finding the Semi-Major and Semi-Minor Axes**:
The relationship between the semi-major axis \( a \) and semi-minor axis \( b \) of the ellipse and its eccentricity \( e \) is given by:
\[ e = \sqrt{1 - \frac{b^2}{a^2}} \]
Rearranging this, we find:
\[ b^2 = a^2(1 - e^2) \]
First, we calculate \( e^2 \):
\[ e^2 = \left(\frac{2\sqrt{2}}{3}\right)^2 = \frac{8}{9} \]
Therefore:
\[ 1 - e^2 = 1 - \frac{8}{9} = \frac{1}{9} \]
Now substituting back:
\[ b^2 = a^2 \cdot \frac{1}{9} \implies b = \frac{a}{3} \]

3. **Calculating the Area of the Ellipse**:
The area \( A \) of the ellipse is given by:
\[ A = \pi a b = \pi a \cdot \frac{a}{3} = \frac{\pi a^2}{3} \]

4. **Calculating the Area of the Circle**:
The area \( A_c \) of the circle is:
\[ A_c = \pi a^2 \]

5. **Finding the Area of the Shaded Region**:
The shaded region (area outside the ellipse but inside the circle) is:
\[ \text{Area of shaded region} = A_c - A = \pi a^2 - \frac{\pi a^2}{3} = \pi a^2 \left(1 - \frac{1}{3}\right) = \pi a^2 \cdot \frac{2}{3} \]

6. **Calculating the Probability \( p \)**:
The probability \( p \) that a point chosen randomly in the circle lies outside the ellipse is given by:
\[ p = \frac{\text{Area of shaded region}}{\text{Area of circle}} = \frac{\frac{2}{3} \pi a^2}{\pi a^2} = \frac{2}{3} \]

7. **Calculating \( 105p \)**:
Now we compute \( 105p \):
\[ 105p = 105 \cdot \frac{2}{3} = 70 \]

### Final Answer:
Thus, the value of \( 105p \) is \( \boxed{70} \).