If the curves `(x^2)/4+y^2=1` and `(x^2)/(a^2)+y^2=1` for a suitable value of `a` cut on four concyclic points, the equation of the circle passing through these four points is

Options

A. `x^(2)+y^(2)=2`
B. `x^(2)+y^(2)=1`
C. `x^(2)+y^(2) =4`
D. none of these

Explanation

To solve the problem, we need to find the equation of the circle passing through the four concyclic points where the given curves intersect. Let's break it down step by step.

### Step 1: Identify the given curves
The curves given are:
1. \(\frac{x^2}{4} + y^2 = 1\) (This is an ellipse)
2. \(\frac{x^2}{a^2} + y^2 = 1\) (This is also an ellipse)

### Step 2: Set up the equations
We can rewrite the equations in a more manageable form:
1. \(x^2 + 4y^2 = 4\) (from the first equation)
2. \(x^2 + a^2y^2 = a^2\) (from the second equation)

### Step 3: Find the points of intersection
To find the points of intersection, we can set the two equations equal to each other. We can express \(y^2\) in terms of \(x^2\) from both equations and equate them.

From the first equation:
\[ y^2 = \frac{4 - x^2}{4} \]

From the second equation:
\[ y^2 = \frac{a^2 - x^2}{a^2} \]

Setting these equal gives:
\[ \frac{4 - x^2}{4} = \frac{a^2 - x^2}{a^2} \]

### Step 4: Cross-multiply and simplify
Cross-multiplying, we get:
\[ (4 - x^2) a^2 = (a^2 - x^2) 4 \]
Expanding both sides:
\[ 4a^2 - a^2x^2 = 4a^2 - 4x^2 \]

### Step 5: Rearranging the equation
Rearranging gives:
\[ -a^2x^2 + 4x^2 = 4a^2 - 4a^2 \]
\[ (4 - a^2)x^2 = 0 \]

### Step 6: Determine the condition for concyclicity
For the curves to intersect at four concyclic points, the coefficient of \(x^2\) must be zero, which means:
\[ 4 - a^2 = 0 \implies a^2 = 4 \implies a = 2 \]

### Step 7: Substitute back to find the circle equation
Now substituting \(a = 2\) back into either of the original equations gives:
1. \(\frac{x^2}{4} + y^2 = 1\)
2. \(\frac{x^2}{4} + y^2 = 1\) (both equations are the same)

### Step 8: Find the equation of the circle
The points of intersection are symmetric about the origin, and they will lie on a circle. The general form of a circle is:
\[ x^2 + y^2 = r^2 \]

Since the maximum value of \(x^2 + y^2\) occurs when \(x^2 + 4y^2 = 4\), we can find the radius:
Setting \(y = 0\) gives \(x^2 = 4\) (points at (2,0) and (-2,0)), and setting \(x = 0\) gives \(4y^2 = 4\) (points at (0,1) and (0,-1)).

Thus, the equation of the circle passing through these points is:
\[ x^2 + y^2 = 2 \]

### Final Answer
The equation of the circle passing through the four concyclic points is:
\[ x^2 + y^2 = 2 \]