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 ...
Show Hint
A line with equal intercepts is x + y = c; its distance from the origin must equal the radius.
Step 1: Geometric View:
The tangent forms an isosceles right triangle with the axes. The perpendicular from the origin to the hypotenuse is the radius 2 and is also the altitude of the triangle.
Step 2: Compute:
In an isosceles right triangle with legs $c$, the altitude to the hypotenuse is $c/\sqrt2$. Setting $c/\sqrt2=2$ gives $c=2\sqrt2$.
Step 3: Answer:
Option (C).
Final Answer:
Option (C).
\[ \boxed{\text{(C) } 2\sqrt{2}} \]