Two circles of radii 9 cm and 2 cm respectively has centres X and 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, |
\[\therefore \]\[X{{Y}^{2}}=X{{Z}^{2}}+Z{{Y}^{2}}\] |
\[\Rightarrow \]\[{{17}^{2}}={{(9+r)}^{2}}+{{(r+2)}^{2}}\] |
\[\Rightarrow \]\[289=81+13r+{{r}^{2}}+{{r}^{2}}+4r+4\] |
\[[\because {{(a+b)}^{2}}={{a}^{2}}+{{b}^{2}}+2ab]\] |
\[\Rightarrow \] \[2{{r}^{2}}+22r-204=0\] |
\[\Rightarrow \] \[{{r}^{2}}+11r-102=0\] |
\[\Rightarrow \] \[{{r}^{2}}+17r-6r-102=0\] |
\[\Rightarrow \]\[r\,\,(r+17)-6\,\,(r+17)=0\] |
\[\Rightarrow \] \[(r-6)(r+17)=0\] |
\[\Rightarrow \] \[r=6\,\,cm\] |
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