The ellipse $4x^2+9y^2=36$  and the hyperbola $a^2{x}^2-y^2=4$  intersect at right angles. Then the equation of the circle through the points of intersection of two conics is

Since ellipse and hyperbola intersect orthogonally, they are confocal Hence, a = 2 (equating foci)
Let the point of intersection in the first quadrant be  $P(x_1,y_1)$.
P lies on the curves. Therefore, $4x_1^2+9y_1^2=36$  and  $4x_1^2+9y_1^2=4$
Adding these two results, we get
$8(x_1^2+y_1^2)=40$ or $x_1^2+y_1^2=5$  or  $r=\sqrt{5}$
Hence, the equation of the circle is  $x^2+y^2=5$