SATHEE: Circle 4 Question 4

Circle 4 Question 4

4. Three circles of radii $a, b, c (a < b < c)$ touch each other externally. If they have $X$ -axis as a common tangent, then

(2019 Main, 9 Jan I)

(a) $a, b, c$ are in $A P$

(b) $\frac{1}{\sqrt{a}} = \frac{1}{\sqrt{b}} + \frac{1}{\sqrt{c}}$

(d) $\frac{1}{\sqrt{b}} = \frac{1}{\sqrt{a}} + \frac{1}{\sqrt{c}}$

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Answer:

Correct Answer: 4. (b)

Solution:

According to given information, we have the following figure.

where $A, B, C$ are the centres of the circles Clearly, $A B = a + b$ (sum of radii) and $B D = b - a$

$∴ A D = \sqrt{(a + b)^{2} - (b - a)^{2}}$

$= 2 \sqrt{a b}$

(using Pythagoras theorem in $△ A B D$ )

Similarly, $A C = a + c$ and $C E = c - a$

$∴$ In $△ A C E, A E = \sqrt{(a + c)^{2} - (c - a)^{2}} = 2 \sqrt{a c}$

Similarly, $B C = b + c$ and $C F = c - b$

$∴ In △ B C F, B F = \sqrt{(b + c)^{2} - (c - b)^{2}} = 2 \sqrt{b c}$

$∵ A D + A E = B F$

$∴ 2 \sqrt{a b} + 2 \sqrt{a c} = 2 \sqrt{b c}$

$\Rightarrow \frac{1}{\sqrt{c}} + \frac{1}{\sqrt{b}} = \frac{1}{\sqrt{a}}$

Circle 4 Question 4

4. Three circles of radii $a, b, c (a < b < c)$ touch each other externally. If they have $X$ -axis as a common tangent, then

Answer:

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Circle 4 Question 4

4. Three circles of radii a,b,c(a<b<c) touch each other externally. If they have X-axis as a common tangent, then

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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4. Three circles of radii $a, b, c (a < b < c)$ touch each other externally. If they have $X$ -axis as a common tangent, then