The radius of the circle passing through the foci of the ellipse `(x^(2))/(16) + (y^(2))/( 9) = 1 ` and having its centre at (0,3) is

Options

A. 4 units
B. 3 units
C. `sqrt(12)` units
D. `(7)/( 2)` units

Answer

To solve the problem of finding the radius of the circle that passes through the foci of the given ellipse and has its center at (0, 3), we can follow these steps:

### Step 1: Identify the parameters of the ellipse
The equation of the ellipse is given as:
\[ \frac{x^2}{16} + \frac{y^2}{9} = 1 \]
From this equation, we can identify:
- \( a^2 = 16 \) so \( a = \sqrt{16} = 4 \)
- \( b^2 = 9 \) so \( b = \sqrt{9} = 3 \)

### Step 2: Calculate the eccentricity of the ellipse
The eccentricity \( e \) of the ellipse is given by the formula:
\[ e = \sqrt{1 - \frac{b^2}{a^2}} \]
Substituting the values of \( b^2 \) and \( a^2 \):
\[ e = \sqrt{1 - \frac{9}{16}} = \sqrt{\frac{16 - 9}{16}} = \sqrt{\frac{7}{16}} = \frac{\sqrt{7}}{4} \]

### Step 3: Find the foci of the ellipse
The foci of the ellipse are located at:
\[ (\pm ae, 0) \]
Substituting the values of \( a \) and \( e \):
\[ (\pm 4 \cdot \frac{\sqrt{7}}{4}, 0) = (\pm \sqrt{7}, 0) \]
Thus, the foci are at the points \( (\sqrt{7}, 0) \) and \( (-\sqrt{7}, 0) \).

### Step 4: Calculate the radius of the circle
The center of the circle is given as \( (0, 3) \). We need to find the radius, which is the distance from the center of the circle to one of the foci. We can use the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Using the focus \( (\sqrt{7}, 0) \) and the center \( (0, 3) \):
\[ d = \sqrt{(\sqrt{7} - 0)^2 + (0 - 3)^2} = \sqrt{(\sqrt{7})^2 + (-3)^2} = \sqrt{7 + 9} = \sqrt{16} = 4 \]

### Conclusion
The radius of the circle is \( 4 \) units.

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