Do the circles x^2 + y^2 = 16 and x^2 + y^2 + 2x + 4y = 36 intersect each other?

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The equation of a circle with center (h, k) and radius r is given by `(x - h)^2 + (y - k)^2 = r^2` .

x^2 + y^2 = 16 has a radius 4 and is centered at the origin.

x^2 + y^2 + 2x + 4y = 36

=> x^2 + 2x + 1 + y^2 + 4y + 4 = 36 + 5 = 41

=> (x + 1)^2 + (y + 2)^2 = 41

The center of this circle is (-1, -2) and the radius is `sqrt 41` .

The distance of (-1, -2) from the origin is `sqrt(1^2 + 2^2) = sqrt 5` . `sqrt 5 + 4 ~~ 6.23` . On the other hand `sqrt 41 ~~ 6.403` . As 6.403 > 6.23, the circle represented by x^2 + y^2 = 16 is completely enclosed within the other circle represented by x^2 + y^2 + 2x + 4y = 36 but they do not touch other.

**The given circles do not intersect each other.**

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