The circle `x^2+y^2=c^2` contains the ellipse `x^2/a^2+y^2/b^2=1` if

Options

A. `c lt a`
B. `c lt b`
C. `c gt` max {a,b}
D. `c gt b`

Answer

To determine the conditions under which the circle \(x^2 + y^2 = c^2\) contains the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), we need to analyze the geometric relationship between the two shapes.

### Step-by-Step Solution:

1. **Understand the Shapes**:
- The circle has a radius \(c\) and is centered at the origin (0,0).
- The ellipse has semi-major axis \(a\) and semi-minor axis \(b\) (assuming \(a \geq b\)).

2. **Identify the Points on the Ellipse**:
- The furthest points on the ellipse from the origin are located at \((\pm a, 0)\) and \((0, \pm b)\).

3. **Condition for the Circle to Contain the Ellipse**:
- For the circle to contain the ellipse, the distance from the origin to the furthest points of the ellipse must be less than or equal to the radius of the circle.
- This gives us two conditions:
- The distance to the points on the x-axis: \(a \leq c\)
- The distance to the points on the y-axis: \(b \leq c\)

4. **Combine the Conditions**:
- Since the circle must contain the ellipse, both conditions must hold true simultaneously. Therefore, we conclude:
\[ c \geq \max(a, b) \]

5. **Final Conclusion**:
- The circle \(x^2 + y^2 = c^2\) contains the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) if:
\[ c > \max(a, b) \]