question_answer

Two circles of radii 9 cm and 2 cm, respectively has centres X, Y and \[\overline{XY}=17\,cm.\] Circle of radius r cm with centre Z touches two given circles externally. If \[\angle XZY=90{}^\circ ,\] then find r.                      [SSC (CGL) 2012]

A) 18 cm              

B) 3 cm

C) 12 cm              

D) 6 cm

Correct Answer: D

Solution :

                    

In \[\Delta XYZ,\] by Pythagoras theorem,

\[\therefore \] \[X{{Y}^{2}}+X{{Z}^{2}}+Z{{Y}^{2}}\]

\[\Rightarrow \] \[{{17}^{2}}={{(9+r)}^{2}}+{{(r+2)}^{2}}\]

\[\Rightarrow \]\[289=81+18r+{{r}^{2}}+{{r}^{2}}+4r+4\]

\[[\because \,\,{{(a+b)}^{2}}={{a}^{2}}+{{b}^{2}}+2ab]\]

\[\Rightarrow \] \[{{r}^{2}}+11r-102=0\]

\[\Rightarrow \] \[{{r}^{2}}+17r-6r-102=0\]

\[\Rightarrow \] \[r\,(r+17)-6\,(r+17)=0\]

\[\Rightarrow \] \[(r-6)(r+17)=0\]\[\Rightarrow \]\[r=6\,cm\]