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Family of Circles

Five circles ...

Question

Five circles $C_{1}, C_{2}, C_{3}, C_{4}, C_{5}$ with radii $r_{1}, r_{2}, r_{3}, r_{4}, r_{5}$ respectively $(r_{1} < r_{2} < r_{3} < r_{4} < r_{5})$ be such that $C_{i}$ and $C_{i} + 1$ touch each other externally for all $i = 1, 2, 3, 4$ . If all the five circles touch two straight lines $L_{1}$ and $L_{2}$ and $r_{1} = 2$ and $r_{5} = 32$ , then $r_{3}$ is (units)

A

8

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B

8.0

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C

8.00

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Solution

Let $∠ O_{1} A T_{1} = x$ , then
$sin x = \frac{r_{2} - r_{1}}{r_{1} + r_{2}}$
Similarly, $sin x = \frac{r_{3} - r_{2}}{r_{3} + r_{2}}$

Now, $\frac{r_{2} - r_{1}}{r_{1} + r_{2}} = \frac{r_{3} - r_{2}}{r_{3} + r_{2}}$
$\Rightarrow r_{2} r_{3} + r_{2}^{2} - r_{1} r_{3} - r_{1} r_{2} = r_{1} r_{3} - r_{1} r_{2} + r_{2} r_{3} - r_{2}^{2} \Rightarrow r_{2}^{2} = r_{1} r_{3}$
So, $r_{1}, r_{2}, r_{3}$ are in G.P.
Similarly, $r_{1}, r_{2}, r_{3}, r_{4}, r_{5}$ are in G.P.
Therefore, $r_{3}^{2} = r_{1} r_{5}$
$\Rightarrow r_{3} = 8$ units

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Similar questions

Q. Five circles

C_{1}, C_{2}, C_{3}, C_{4}, C_{5}

r_{1}, r_{2}, r_{3}, r_{4}, r_{5}

(r_{1} < r_{2} < r_{3} < r_{4} < r_{5})

C_{i}

C_{i} + 1

touch each other externally for all

i = 1, 2, 3, 4

. If all the five circles touches two straight lines

L_{1}

L_{2}

r_{1} = 2

r_{5} = 32

r_{3}

Q. There are three circles

C_{1}, C_{2}, C_{3}

r_{1}, r_{2}, r_{3} (r_{1} < r_{2} < r_{3})

respectively touching each other externally as shown in the figure and

L_{1}, L_{2}

are two common tangents to these circles from point

P

Which of the following is(are) correct?

Q. Three circles with centres

C_{1}, C_{2}, C_{3}

r_{1}, r_{2}, r_{3}

r_{1} < r_{2} < r_{3}

are placed as shown in the figure.
Then

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n \geq 3

C_{1}, C_{2}, . . . . C_{n}

, be circles with radii

r_{1}, r_{2}, . . . . . r_{n}

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C_{i + 1}

touch externally for

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. It is also given that the x-axis and the line

y = 2 \sqrt{2} x + 10

are tangential to each of the circles. Then

r_{1}, r_{2}, . . . . ., r_{n}

Q. Three circles of radii

r_{1}

r_{2}

r_{3}

are drawn concentric to each other. The radii

r_{1}

r_{2}

are such that the area of the circle with radius

r_{1}

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