A line making equal intercepts on coordinate axes and is tangent to the circle $x^2 + y^2 = 4$. The length of each intercept made by line on the coordinate axes is ...

Let the equation of the line making equal intercepts on the coordinate axes be $\dfrac{x}{a} + \dfrac{y}{a} = 1$.

This can be written as $x + y - a = 0$.

The line is tangent to the circle $x^2 + y^2 = 4$.

The center of the circle is $(0, 0)$ and its radius is $2$.

For a line to be tangent to a circle, the perpendicular distance from the center of the circle to the line must be equal to the radius.

$\dfrac{|0 + 0 - a|}{\sqrt{1^2 + 1^2}} = 2$

$\dfrac{|a|}{\sqrt{2}} = 2$

$|a| = 2\sqrt{2}$

The length of each intercept made by the line on the coordinate axes is $2\sqrt{2}$.