Question

If $\left( m_{i},\frac{1}{m_{i}} \right),i = 1,2,3,4$ are concylic points, then the value of $m_{1}.m_{2}.m_{3}.m_{4}$ is

• A. 1
• B. – 1
• C. 0
• D. None of these

Answer 1

Answer: (A)

1

Answer 2

Let the equation of circle be $x^{2} + y^{2} + 2gx + 2fy + c = 0$Since the point $\left( m_{i},\frac{1}{m_{i}} \right)$ lies on this circle

$\therefore$ ${m_{i}}^{2} + \frac{1}{{m_{i}}^{2}} + 2gm_{i} + \frac{2f}{m_{i}} + c = 0$

$\Rightarrow$ ${m_{i}}^{4} + 2g{m_{i}}^{3} + c{m_{i}}^{2} + 2fm_{i} + 1 = 0$Clearly its roots are $m_{1},m_{2},m_{3}$ and $m_{4}$,

$\therefore$ $m_{1}.m_{2}.m_{3}.m_{4}$ = product of roots $= \frac{1}{1} = 1$