Let C 1 , C 2 , C 3 are three circles with radius r 1 , r 2 , r 3 ( r 1

Question

Let C 1 ​ , C 2 ​ , C 3 ​ are three circles with radius r 1 ​ , r 2 ​ , r 3 ​ ( r 1 ​ < r 2 ​ < r 3 ​ ) and touches each other externally. Also let L 1 ​ and L 2 ​ are two direct common tangents to three circles. Let point of intersection of these lines is P and D, E, F are points of contact of L 2 ​ with C 1 ​ , C 2 ​ , C 3 ​ . Match the following. List-I List-II (A) r 3 ​ equals. (p) 2 r 2 ​ ​ ( r 1 ​ ​ + r 3 ​ ​ ) (B) The distance DE is equal to. (q) r 2 ​ − r 1 ​ 2 r 1 3/2 ​ ⋅ r 2 1/2 ​ ​ (C) The distance PD can be given as. (r) r 1 ​ r 2 2 ​ ​ (D) The distance DF is equal to. (s) 2 r 1 ​ r 2 ​ ​ (t) r 1 ​ r 2 2 ​ − r 1 2 ​ ​

TestHub · Accepted Answer

D E = ( r 1 ​ + r 2 ​ ) 2 − ( r 2 ​ − r 1 ​ ) 2 ​ = 2 r 1 ​ r 2 ​ ​ → B-s; similarly EF = 2 r 2 ​ r 3 ​ ​ . Let P D = x , ∠ A P D = θ : tan θ = x r 1 ​ ​ = x + 2 r 1 ​ r 2 ​ ​ r 2 ​ ​ = x + 2 r 1 ​ r 2 ​ ​ + 2 r 2 ​ r 3 ​ ​ r 3 ​ ​ . Hence 2 r 1 ​ r 2 ​ ​ r 2 ​ − r 1 ​ ​ = 2 r 2 ​ r 3 ​ ​ r 3 ​ − r 2 ​ ​ ⇒ r 3 ​ ​ ( r 2 ​ − r 1 ​ ) = r 1 ​ ​ ( r 3 ​ − r 2 ​ ) ⇒ r 2 ​ ( r 1 ​ ​ + r 3 ​ ​ ) = r 1 ​ r 3 ​ ​ ( r 1 ​ ​ + r 3 ​ ​ ) ⇒ r 2 2 ​ = r 1 ​ r 3 ​ ⇒ r 3 ​ = r 1 ​ r 2 2 ​ ​ → A-R. Also x = r 2 ​ − r 1 ​ r 1 ​ ⋅ 2 r 1 ​ r 2 ​ ​ ​ = r 2 ​ − r 1 ​ 2 r 1 3/2 ​ r 2 1/2 ​ ​ → C − Q . And D F = D E + EF = 2 r 1 ​ r 2 ​ ​ + 2 r 2 ​ r 3 ​ ​ = 2 r 2 ​ ​ ( r 1 ​ ​ + r 3 ​ ​ ) → D − P .

List-I	List-II
(A) $r_{3}$ equals.	(p) $2 r_{2} (r_{1} + r_{3})$
(B) The distance DE is equal to.	(q) $\frac{2 r _{1}^{3/2} \cdot r _{2}^{1/2}}{r _{2} - r _{1}}$
(C) The distance PD can be given as.	(r) $\frac{r _{2}^{2}}{r _{1}}$
(D) The distance DF is equal to.	(s) $2 r_{1} r_{2}$
	(t) $\frac{r _{2}^{2} - r _{1}^{2}}{r _{1}}$

Mathematics - Circle Question with Solution | TestHub

Options:

Doubts & Discussion