Let E be the ellipse `(x^(2))/(9)+(y^(2))/(4)=1` and C be the circle `x^(2)+y^(2)=9`. If P = (1,2), then which one of the following is correct?

Options

A. Q lies inside C but outside E
B. Q lies outside both C and E
C. P lies inside both C and E
D. P lies inside C but outside E

Answer

To solve the problem, we need to determine the position of the point P = (1, 2) in relation to the given ellipse E and circle C.

### Step-by-Step Solution:

1. **Identify the equations of the ellipse and the circle:**
- The equation of the ellipse \( E \) is given by:
\[ \frac{x^2}{9} + \frac{y^2}{4} = 1 \]
- The equation of the circle \( C \) is given by:
\[ x^2 + y^2 = 9 \]

2. **Substitute the coordinates of point P into the circle's equation:**
- The coordinates of point P are \( (1, 2) \).
- Substitute \( x = 1 \) and \( y = 2 \) into the circle's equation:
\[ 1^2 + 2^2 = 1 + 4 = 5 \]
- Since \( 5 < 9 \), point P lies **inside the circle**.

3. **Substitute the coordinates of point P into the ellipse's equation:**
- Substitute \( x = 1 \) and \( y = 2 \) into the ellipse's equation:
\[ \frac{1^2}{9} + \frac{2^2}{4} = \frac{1}{9} + \frac{4}{4} = \frac{1}{9} + 1 = \frac{1}{9} + \frac{9}{9} = \frac{10}{9} \]
- Since \( \frac{10}{9} > 1 \), point P lies **outside the ellipse**.

4. **Conclusion:**
- Point P lies inside the circle and outside the ellipse. Therefore, the correct statement is:
- Point P lies inside circle C but outside ellipse E.