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 point of intersection of two conic is

Since ellipse and hyperbola intersect orthogonally, they are confocal hence $a=2$
Let  $P(x_1,y_1)$ be the point of intersection of both the curves therefore,  $4x_1^2+9y_1^2=36$ and  $4x_1^2-y_1^2=4$
$\Rightarrow x_1^2+y_1^2=5$
Hence the equation of circle is  $x^2+y^2=5$