Question

Let there be $n\ge 3$ circles in a plane. The value of n for which the number of radical centres is equal to the number of radical axes is (assume that all radical axes and radical centre exist and are different)
(a) 7
(b) 6
(c) 5
(d) none of these

Answer 1

The radical center of three circles is the intersection point of the three radical axes of the pairs of circles. So for a radical center we need 3 circles and for a radical axis we need 2 circles. Using a combination formula we will solve this problem.

Complete step-by-step answer:
It is mentioned in the question that the number of circles is n. Also $n\ge 3$.
Now for a radical center we need 3 circles. Using this information we get,
The total number of radical centers $={}^{n}{{C}_{3}}.....(1)$
Also for a radical axis 2 circles are required. Using this information we get,
The total number of radical axis $={}^{n}{{C}_{2}}.....(2)$
Now we need to find the value of n for which the number of radical centres is equal to the number of radical axes. Hence using this information we equate equation (1) and equation (2) and so we get,
$\Rightarrow {}^{n}{{C}_{3}}={}^{n}{{C}_{2}}.....(3)$
Now applying the complementary combination formula ${}^{n}{{C}_{r}}={}^{n}{{C}_{n-r}}$ in left hand side of equation (3), we get,
$\Rightarrow {}^{n}{{C}_{n-3}}={}^{n}{{C}_{2}}.....(3)$
Now equating the terms in equation (3) we get,
$\Rightarrow n-3=2.....(4)$
Now isolating n in equation (4) and solving we get,
$\Rightarrow n=3+2=5$
Hence the value of n is 5. So the correct answer is option (c).

Note: A combination is a way to order or arrange a set or number of things uniquely. Here remembering the definition of radical center and radical axis is the key. Also the complementary property of combination ${}^{n}{{C}_{r}}={}^{n}{{C}_{n-r}}$ is important.