isi-entrance 2011 Q11

isi-entrance · India · solved Circles Circles Tangent to Each Other or to Axes

Let $C _ { 1 } , C _ { 2 }$ and $C _ { 3 }$ be three circles lying in the same quadrant, each touching both the axes. Suppose also that $C _ { 1 }$ touches $C _ { 2 }$ and $C _ { 2 }$ touches $C _ { 3 }$. If the area of the smallest circle is 1 unit, then area of the largest circle is
(a) $\{ ( \sqrt{2} + 1 ) / ( \sqrt{2} - 1 ) \} ^ { 4 }$
(b) $( 1 + \sqrt{2} ) ^ { 2 }$
(c) $( 2 + \sqrt{2} ) ^ { 2 }$
(d) $2 ^ { 4 }$

(A) Let radii be $r < r_1 < r_2$. From the touching condition: $r_1/r = (\sqrt{2}+1)/(\sqrt{2}-1)$ and $r_2/r_1 = (\sqrt{2}+1)/(\sqrt{2}-1)$. So $r_2/r = \{(\sqrt{2}+1)/(\sqrt{2}-1)\}^2$, and area of largest circle $= \{(\sqrt{2}+1)/(\sqrt{2}-1)\}^4$ (since $\pi r^2 = 1$).

Let $C _ { 1 } , C _ { 2 }$ and $C _ { 3 }$ be three circles lying in the same quadrant, each touching both the axes. Suppose also that $C _ { 1 }$ touches $C _ { 2 }$ and $C _ { 2 }$ touches $C _ { 3 }$. If the area of the smallest circle is 1 unit, then area of the largest circle is\\
(a) $\{ ( \sqrt{2} + 1 ) / ( \sqrt{2} - 1 ) \} ^ { 4 }$\\
(b) $( 1 + \sqrt{2} ) ^ { 2 }$\\
(c) $( 2 + \sqrt{2} ) ^ { 2 }$\\
(d) $2 ^ { 4 }$

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