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

Hint: 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.