Let `C : x^(2) + y^(2) = 9, E : (x^(2))/(9) + (y^(2))/(4) =1` and L : y = 2x be three curves. IF R is the point of intersection of the line L with the line `x =1 , then

Options

A. (a)R lies inside both C and E
B. (b)R lies outside both C and E
C. (c)R lies on both C and E
D. (d)R lies inside C but outside E

Answer

To solve the problem, we need to analyze the given curves and determine the position of the point R (1, 2) relative to the circle C and the ellipse E.

### Step-by-Step Solution:

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

2. **Find the Point of Intersection R:**
- We need to find the point of intersection of the line \( L \) with the line \( x = 1 \).
- Substitute \( x = 1 \) into the equation of line \( L \):
\[ y = 2(1) = 2 \]
- Therefore, the point \( R \) is:
\[ R(1, 2) \]

3. **Check if R lies inside or outside the Circle C:**
- To determine if point \( R(1, 2) \) lies inside, outside, or on the circle \( C \), we substitute \( R \) into the equation of the circle:
\[ 1^2 + 2^2 = 1 + 4 = 5 \]
- Since \( 5 < 9 \) (the right side of the circle's equation), point \( R \) lies **inside** the circle \( C \).

4. **Check if R lies inside or outside the Ellipse E:**
- Now, we substitute \( R(1, 2) \) into the equation of the ellipse \( E \):
\[ \frac{1^2}{9} + \frac{2^2}{4} = \frac{1}{9} + \frac{4}{4} = \frac{1}{9} + 1 = \frac{1}{9} + \frac{9}{9} = \frac{10}{9} \]
- Since \( \frac{10}{9} > 1 \) (the right side of the ellipse's equation), point \( R \) lies **outside** the ellipse \( E \).

5. **Conclusion:**
- The point \( R(1, 2) \) is **inside** the circle \( C \) and **outside** the ellipse \( E \).

### Final Answer:
The point \( R(1, 2) \) is inside the circle \( C \) but outside the ellipse \( E \).