Circle And System Of Circles Question 250
Question: The radical centre of three circles described on the three sides of a triangle as diameter is
[EAMCET 1994]
Options:
A) The orthocentre
B) The circumcentre
C) The incentre of the triangle
D) The centroid
Show Answer
Answer:
Correct Answer: A
Solution:
Let us consider a triangle as shown in fig. Equations of the circles with AB, BC and CA as diameters are $ S_1\equiv (x+a)(x-a)+y^{2}=0 $
$ S_2\equiv (x-a)(x-\alpha )+y(y-\beta )=0 $ and $ S_3\equiv (x+a)(x-\alpha )+y(y-\beta )=0 $ i.e., $ S_1\equiv x^{2}+y^{2}-a^{2}=0 $
$ S_2\equiv x^{2}+y^{2}-(a+\alpha )x-\beta y+a\alpha =0 $ and $ S_3\equiv x^{2}+y^{2}-(\alpha -a)x-\beta y-a\alpha =0 $
Radical axis of $ S_2 $ and $ S_3 $ is $ S_3-S_2=0 $ i.e., $ 2ax-2a\alpha =0 $
therefore $ 2a(x-\alpha )=0 $ , as $ a\ne 0 $ , $ x=\alpha $
But $ x=\alpha $ is the orthogonal through C.
Similarly other radical axes will be orthogonals through A and B.
Hence radical centre will be the orthocentre.
BETA
