CAT 2024 Slot 2 QA Question 16: Three circles of equal radii touch (but not cross) each other externally. Two other circles, $x$ and $y$, are drawn such that both touch (but not cross) each of the three previous circles. If the radi

Question

Three circles of equal radii touch (but not cross) each other externally. Two other circles, $x$ and $y$, are drawn such that both touch (but not cross) each of the three previous circles. If the radi

Accepted Answer

Let the radius of each of the three equal circles be $r$. Use Descartes’ circle theorem in terms of curvature, where curvature is the reciprocal of radius.
For three mutually tangent equal circles, the two circles tangent to all three have signed curvatures:
$k = \dfrac{3}{r} \pm \dfrac{2\sqrt{3}}{r

Three circles of equal radii touch (but not cross) each other externally. Two other circles, xxx and yyy, are drawn such that both touch (but not cross) each of the three previous circles. If the radius of xxx is more than that of yyy, the ratio of the radii of x and y is

Three circles of equal radii touch (but not cross) each other externally. Two other circles, $x$ and $y$ , are drawn such that both touch (but not cross) each of the three previous circles. If the radius of $x$ is more than that of $y$ , the ratio of the radii of x and y is