Let circles $S_{1}$ and $S_{2}$ of radii $r_{1}$ and $r_{2} (r_{1} > r_{2})$ respectively touches each other externally. Circle S of radius r touches $S_{1}$ and $S_{2}$ externally and also their direct common tangent. If the largest angle of triangle formed by joining centre of $S_{1}, S_{2}$ and S is A , then which of the following option(s) is / are true?

see full answer

Start JEE / NEET / Foundation preparation at rupees 99/day !!

21% of IItians & 23% of AIIMS delhi doctors are from Sri Chaitanya institute !!

An Intiative by Sri Chaitanya

$\tan \frac{A}{2} = \frac{4}{3}$ if $r_{1} = r_{2}$

$A = 90^{°}$ if $r_{1} = r_{2}$

A can be less than $90^{°}$

A is greater than $90^{°}$

answer is A, C.

(Unlock A.I Detailed Solution for FREE)

Ready to Test Your Skills?

Check your Performance Today with our Free Mock Test used by Toppers!

Take Free Test

Detailed Solution

Since radii are in H.P.
$r_{1}^{2} + r_{2}^{2} + 2 r_{1} r_{2} - r^{2} - r_{1}^{2} - 2 {rr}_{1} - r^{2} - r_{2}^{2} - 2 {rr}_{2}$
$= - 2 {rr}_{1} - 2 {rr}_{2} + 2 r_{1} r_{2} - 2 r^{2} = 4 r \sqrt{r_{1} r_{2}} - 2 r^{2}$
$= 2 r (2 \sqrt{r_{1} r_{2}} - r) = positive \Rightarrow Δ C_{1} {CC}_{2}$ is obtuse angle triangle
If $r_{1} = r_{2}$ then from equation (1), $r_{1} = 4 r$ and $\sin \frac{A}{2} = \frac{r_{1}}{r + r_{1}}$