Get Answers to all your Questions

header-bg qa

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

Option: 1

8


Option: 2

16


Option: 3

4


Option: 4

32


Answers (1)

best_answer

Let P, Q and R be the centre of the circles and O the point of intersection of the common tangents at A, B and C.

\mathrm{\text { We know that } \Delta=\sqrt{s(s-a)(s-b)(s-c)}}

\mathrm{\text { Area of the triangle, } P Q R=\sqrt{\left(r_1+r_2+r_3\right) r_1 \cdot r_2 \cdot r_3}}     ...[1]

\mathrm{\text { Also, area of } \triangle P Q R=\frac{1}{2}(Q R \times 4)+\frac{1}{2}(R P \times 4)+\frac{1}{2}(P Q \times 4)}

\mathrm{\begin{aligned} & (\text { Since } O A=O B=O C=4) \\ & =2(Q R+R P+P Q)=4\left(r_1+r_2+r_3\right) \end{aligned}}       ...[2]

Now, from (1) and (2)

\mathrm{\begin{aligned} & 4\left(r_1+r_2+r_3\right)=\sqrt{\left(r_1+r_2+r_3\right) r_1 \cdot r_2 \cdot r_3} \\ & \Rightarrow 16\left(r_1+r_2+r_3\right)^2=\left(r_1+r_2+r_3\right) r_1 \cdot r_2 \cdot r_3 \\ & \Rightarrow \frac{r_1 \cdot r_2 \cdot r_3}{r_1+r_2+r_3}=16 \end{aligned}}

Posted by

sudhir.kumar

View full answer

JEE Main high-scoring chapters and topics

Study 40% syllabus and score up to 100% marks in JEE