Q113 / 200 · Manipal (MET) 2010 · Mathematics

If $\left(m_i, \frac{1}{m_i}\right)$ are four distinct points on a circle, then

  • Correct answer
    A
    $m_1 m_2 m_3 m_4=1$
  • B
    $m_1 m_2 m_3 m_4=-1$
  • C
    $m_1 m_2 m_3 m_4=\frac{1}{2}$
  • D
    $\frac{1}{m_1}+\frac{1}{m_2}+\frac{1}{m_3}+\frac{1}{m_4}=\frac{1}{4}$

Solution

Let the points $\left(m_i, \frac{1}{m_i}\right), i=1,2,3,4$ lie on the circle $x^2+y^2+2 g x+2 f y+c=0$ Then, $\Rightarrow m_i^2+\frac{1}{m_i^2}+2 g m_i+\frac{2 f}{m_i}+c=0, i=1,2,3,4$ $\Rightarrow m_i^4+2 g m_i^3+c m_i^2+2 f m_i+1=0$, $i=1,2,3,4$ $\Rightarrow m_1, m_2, m_3$ and $m_4$ are the roots of the equatión $m^4+2 g m^3+c m^2+2 f m+1=0$ $\Rightarrow m_1 m_2 m_3 m_4=\frac{1}{1}=1$

Hence, the correct answer is option: 'A'.

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