Quantitative Aptitude Ques 1282

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

Options:

A) 18 cm

B) 3 cm

C) 12 cm

D) 6 cm

Show Answer

Answer:

Correct Answer: D

Solution:

  • In $ \Delta XYZ, $ By Pythagoras theorem,

$ \therefore $ $ XY^{2}=XZ^{2}+ZY^{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 $ $ 2r^{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=6cm $

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