The ellipse `4x^2+9y^2=36` and the hyperbola `a^2x^2-y^2=4` intersect at right angles. Then the equation of the circle through the points of intersection of two conics is (a) `x^2+y^2=5` (b) `sqrt(5)(x^2+y^2)-3x-4y=0` (c) `sqrt(5)(x^2+y^2)+3x+4y=0` (d) `x^2+y^2=25`

Options

A. `x^(2)+y^(2)=5`
B. `sqrt5(x^(2)+y^(2))-3x-4y=0`
C. `sqrt5(x^(2)+y^(2))+3x+4y=0`
D. `x^(2)+y^(2)=25`

Explanation

To solve the problem, we need to find the equation of the circle that passes through the points of intersection of the given ellipse and hyperbola, which intersect at right angles.

### Step-by-Step Solution:

1. **Write the equations of the conics**:
- The equation of the ellipse is given as:
\[ 4x^2 + 9y^2 = 36 \]
- The equation of the hyperbola is given as:
\[ a^2x^2 - y^2 = 4 \]

2. **Convert the equations to standard form**:
- For the ellipse:
\[ \frac{x^2}{9} + \frac{y^2}{4} = 1 \]
Here, \( a^2 = 9 \) and \( b^2 = 4 \).
- For the hyperbola:
\[ \frac{x^2}{\frac{4}{a^2}} - \frac{y^2}{4} = 1 \]

3. **Condition for orthogonality**:
- The ellipse and hyperbola intersect orthogonally if the product of their slopes at the points of intersection is -1. This condition implies that the foci of the ellipse and hyperbola are the same (confocal).

4. **Find the foci**:
- For the ellipse, the foci are located at \( (\pm c, 0) \) where \( c = \sqrt{a^2 - b^2} = \sqrt{9 - 4} = \sqrt{5} \).
- For the hyperbola, the foci are also located at \( (\pm c, 0) \) where \( c = \sqrt{a^2 + b^2} \).

5. **Set the foci equal**:
- Since they are confocal, we set the foci equal:
\[ \sqrt{5} = \sqrt{a^2 + 4} \]
- Squaring both sides gives:
\[ 5 = a^2 + 4 \implies a^2 = 1 \]

6. **Substitute \( a^2 \) back into the hyperbola equation**:
- The hyperbola becomes:
\[ x^2 - y^2 = 4 \]

7. **Find points of intersection**:
- We will solve the system of equations:
1. \( 4x^2 + 9y^2 = 36 \)
2. \( x^2 - y^2 = 4 \)

8. **Multiply the hyperbola equation by 4**:
- This gives:
\[ 4x^2 - 4y^2 = 16 \]

9. **Add the two equations**:
- Adding gives:
\[ 4x^2 + 9y^2 + 4x^2 - 4y^2 = 36 + 16 \]
\[ 8x^2 + 5y^2 = 52 \]
\[ 8x^2 + 5y^2 = 52 \implies 8x^2 + 5y^2 = 52 \]

10. **Solve for \( x^2 + y^2 \)**:
- Rearranging gives:
\[ x^2 + y^2 = 5 \]

11. **Equation of the circle**:
- The equation of the circle through the points of intersection is:
\[ x^2 + y^2 = 5 \]

### Conclusion:
The equation of the circle through the points of intersection of the ellipse and hyperbola is:
\[ \boxed{x^2 + y^2 = 5} \]