The equation of the circle drawn with the two foci of `(x^(2))/(a^(2))+(y^(2))/(b^(2))=1 ` as the end -point of a diameter , is

Options

A. `x^(2)+y^(2)=a^(2)+b^(2)`
B. `x^(2)+y^(2)=a^(2) `
C. `x^(2)+y^(2)=2a^(2) `
D. `x^(2)+y^(2)=a^(2)-b^(2) `

Explanation

To find the equation of the circle drawn with the two foci of the ellipse \((\frac{x^2}{a^2}) + (\frac{y^2}{b^2}) = 1\) as the endpoints of a diameter, we can follow these steps:

### Step 1: Identify the foci of the ellipse
The foci of the ellipse given by the equation \((\frac{x^2}{a^2}) + (\frac{y^2}{b^2}) = 1\) are located at the points \((\pm c, 0)\), where \(c\) is defined as:
\[ c = \sqrt{a^2 - b^2} \]

### Step 2: Determine the coordinates of the foci
From the formula for \(c\), the coordinates of the foci are:
\[ F_1 = (c, 0) \quad \text{and} \quad F_2 = (-c, 0) \]
Substituting for \(c\):
\[ F_1 = \left(\sqrt{a^2 - b^2}, 0\right) \quad \text{and} \quad F_2 = \left(-\sqrt{a^2 - b^2}, 0\right) \]

### Step 3: Calculate the distance between the foci
The distance \(d\) between the two foci \(F_1\) and \(F_2\) is:
\[ d = F_1 - F_2 = \sqrt{a^2 - b^2} - (-\sqrt{a^2 - b^2}) = 2\sqrt{a^2 - b^2} \]

### Step 4: Find the radius of the circle
The radius \(R\) of the circle whose diameter is the distance between the foci is half of that distance:
\[ R = \frac{d}{2} = \frac{2\sqrt{a^2 - b^2}}{2} = \sqrt{a^2 - b^2} \]

### Step 5: Write the equation of the circle
The equation of a circle with center at the origin \((0, 0)\) and radius \(R\) is given by:
\[ x^2 + y^2 = R^2 \]
Substituting for \(R\):
\[ x^2 + y^2 = (\sqrt{a^2 - b^2})^2 \]
Thus, the equation simplifies to:
\[ x^2 + y^2 = a^2 - b^2 \]

### Final Answer
The equation of the circle is:
\[ x^2 + y^2 = a^2 - b^2 \]