SATHEE: Circle 2 Question 22

Circle 2 Question 22

23. Three circles touch one another externally. The tangents at their points of contact meet at a point whose distance from a point of contact is 4 . Find the ratio of the product of the radii to the sum of the radii of the circles.

$(1992, 5$ M)

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Solution:

Suppose the circles have centres at $C_{1}, C_{2}$ and $C_{3}$ with radius $R_{1}, R_{2}$ and $R_{3}$ , respectively. Let the circles touch at $A, B$ and $C$ . Let the common tangents at $A, B$ and $C$ meet at $O$ . We have, $O A = O B = O C = 4$ [given]. Now, the circle with centre at $O$ and passing through $A, B$ and $C$ is the incircle of the triangle $C_{1} C_{2} C_{3}$ (because $O A ⊥ C_{1} C_{2}$ ).

Therefore, the inradius of $Δ C_{1} C_{2} C_{3}$ is 4 .

and

$r = \frac{Δ}{s}$

Now, perimeter of a triangle

$\begin{aligned} 2 s = R_{1} + R_{2} + R_{2} + R_{3} + R_{3} + R_{1} \\ \Rightarrow 2 s = 2 (R_{1} + R_{2} + R_{3}) \\ \Rightarrow s = R_{1} + R_{2} + R_{3} \\ and Δ = \sqrt{s (s - a) (s - b) (s - c)} \\ = \sqrt{(R_{1} + R_{2} + R_{3}) (R_{3}) (R_{2}) (R_{1})} \\ From Eq. (i), 4 = \frac{\sqrt{R_{1} R_{2} R_{3} (R_{1} + R_{2} + R_{3})}}{R_{1} + R_{2} + R_{3}} \\ \begin{aligned} \Rightarrow & 16 & = \frac{R_{1} R_{2} R_{3} (R_{1} + R_{2} + R_{3})}{{(R_{1} + R_{2} + R_{3})}^{2}} \\ \Rightarrow & 16 & = \frac{R_{1} R_{2} R_{3}}{R_{1} + R_{2} + R_{3}} \end{aligned} \end{aligned}$

Circle 2 Question 22

23. Three circles touch one another externally. The tangents at their points of contact meet at a point whose distance from a point of contact is 4 . Find the ratio of the product of the radii to the sum of the radii of the circles.

Solution:

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Circle 2 Question 22

23. Three circles touch one another externally. The tangents at their points of contact meet at a point whose distance from a point of contact is 4 . Find the ratio of the product of the radii to the sum of the radii of the circles.

Solution:

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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