isi-entrance None Q8

isi-entrance · India · subjective_collection Geometric Sequences and Series Fractal/Iterative Geometric Construction (Area, Length, or Perimeter Series)

Let $\{C_n\}$ be an infinite sequence of circles lying in the positive quadrant of the $XY$-plane, with strictly decreasing radii and satisfying the following conditions. Each $C_n$ touches both $X$-axis and the $Y$-axis. Further, for all $n \geq 1$, the circle $C_{n+1}$ touches the circle $C_n$ externally. If $C_1$ has radius 10 cm, then show that the sum of the areas of all these circles is $\frac{25\pi}{3\sqrt{2}-4}$ sq. cm.

Since each circle touches both axes, the centre of $C_n$ with radius $R_n$ is at $(R_n, R_n)$.
$OO_1 = R_1\sqrt{2}$, and $R_2 = R_1 \cdot \frac{\sqrt{2}-1}{\sqrt{2}+1}$, $R_3 = R_1\left(\frac{\sqrt{2}-1}{\sqrt{2}+1}\right)^2$, etc.
Let $r = \left(\frac{\sqrt{2}-1}{\sqrt{2}+1}\right)^2$.
$$\text{Total Area} = \pi(R_1^2 + R_2^2 + \cdots) = \pi R_1^2 \cdot \frac{1}{1-r} = \pi R_1^2 \cdot \frac{3+2\sqrt{2}}{4\sqrt{2}} = \pi R_1^2 \cdot \frac{3\sqrt{2}+4}{8}$$
$$= \frac{\pi}{8} R_1^2 \cdot \frac{18-16}{3\sqrt{2}-4} = \frac{\pi}{4} \cdot 100 \cdot \frac{1}{3\sqrt{2}-4} = \frac{25\pi}{3\sqrt{2}-4} \text{ sq.cm.}$$

Let $\{C_n\}$ be an infinite sequence of circles lying in the positive quadrant of the $XY$-plane, with strictly decreasing radii and satisfying the following conditions. Each $C_n$ touches both $X$-axis and the $Y$-axis. Further, for all $n \geq 1$, the circle $C_{n+1}$ touches the circle $C_n$ externally. If $C_1$ has radius 10 cm, then show that the sum of the areas of all these circles is $\frac{25\pi}{3\sqrt{2}-4}$ sq. cm.

Paper Questions

Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10