A circle is drawn with the two foci of an ellipse `(x^(2))/(a^(2)) +(y^(2))/(b^(2))=1` at the end of diameter. What is the equation of the circle?

Options

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

Answer

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

### Step 1: Identify the foci of the ellipse
The foci of the ellipse are located at \((\pm ae, 0)\), where \(e\) is the eccentricity of the ellipse. The eccentricity \(e\) is given by the formula:
\[ e = \sqrt{1 - \frac{b^2}{a^2}} \]
Thus, the coordinates of the foci are:
\[ F_1 = (ae, 0) \quad \text{and} \quad F_2 = (-ae, 0) \]

### Step 2: Determine the center of the circle
The center of the circle will be the midpoint of the line segment joining the two foci. The midpoint \(M\) can be calculated as:
\[ M = \left( \frac{ae + (-ae)}{2}, \frac{0 + 0}{2} \right) = (0, 0) \]
So, the center of the circle is at the origin \((0, 0)\).

### Step 3: Calculate the radius of the circle
The radius of the circle is the distance from the center to either of the foci. We can calculate the distance from the origin to one of the foci, say \(F_1\):
\[ \text{Radius} = \sqrt{(ae - 0)^2 + (0 - 0)^2} = ae \]

### Step 4: Write the equation of the circle
The standard equation of a circle with center \((h, k)\) and radius \(r\) is given by:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
Substituting \(h = 0\), \(k = 0\), and \(r = ae\), we get:
\[ x^2 + y^2 = (ae)^2 \]

### Step 5: Substitute the value of \(e\)
Now, substituting \(e = \sqrt{1 - \frac{b^2}{a^2}}\) into the equation:
\[ x^2 + y^2 = a^2 e^2 = a^2 \left(1 - \frac{b^2}{a^2}\right) = a^2 - b^2 \]

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