If (mi,1mi), i = 1, 2, 3, 4 are con - cyclic points, then the value of m1m2m3m4

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If

(m_{i}, \frac{1}{m_{i}})

, i = 1, 2, 3, 4 are con - cyclic points, then the value of

m_{1} m_{2} m_{3} m_{4}

is

Options:

A . 1

B . -1

C . 0

D . 2

Answer: Option A
:
A
Let equation of circle be

x^{2} + y^{2}

+ 2gx + 2fy + c = 0. If

(m, \frac{1}{m})

lies on this circle, then

m^{2} + \frac{1}{m^{2}} + 2 g m + 2 f \frac{1}{m} + c = 0

or

m^{4} + 2 g m^{3} + 2 f m + c m^{2} + 1 = 0

This is a fourth degree equation in m having

m_{1}, m_{2}, m_{3}, m_{4}

as its roots.
Therefore,

m_{1} m_{2} m_{3} m_{4}

= product of roots =

\frac{1}{1}

= 1.

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