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, |
\[\therefore \] \[X{{Y}^{2}}+X{{Z}^{2}}+Z{{Y}^{2}}\] |
\[\Rightarrow \] \[{{17}^{2}}={{(9+r)}^{2}}+{{(r+2)}^{2}}\] |
\[\Rightarrow \]\[289=81+18r+{{r}^{2}}+{{r}^{2}}+4r+4\] |
\[[\because \,\,{{(a+b)}^{2}}={{a}^{2}}+{{b}^{2}}+2ab]\] |
\[\Rightarrow \] \[{{r}^{2}}+11r-102=0\] |
\[\Rightarrow \] \[{{r}^{2}}+17r-6r-102=0\] |
\[\Rightarrow \] \[r\,(r+17)-6\,(r+17)=0\] |
\[\Rightarrow \] \[(r-6)(r+17)=0\]\[\Rightarrow \]\[r=6\,cm\] |
You need to login to perform this action.
You will be redirected in
3 sec