Circle 1 Question 20
20. If $\left(m _i, 1 / m _i\right), m _i>0, i=1,2,3,4$ are four distinct points on a circle, then show that $m _1 m _2 m _3 m _4=1$.
$(1989,2 M)$
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Solution:
- Let the points $m _i, \frac{1}{m _i} ; i=1,2,3,4$ lie on a circle $x^{2}+y^{2}+2 g x+2 f y+c=0$.
Then, $m _i^{2}+\frac{1}{m _i^{2}}+2 g m _i+\frac{2 f}{m _i}+c=0$;
Since, $m _i^{4}+2 g m _i^{3}+c m _i^{2}+2 f m _i+1=0 ; i=1,2,3,4$ $\Rightarrow m _1, m _2, m _3$ and $m _4$ are the roots of the equation
$$ \begin{array}{rlrl} & m^{4}+2 gm^{3}+c m^{2}+2 f m+1 & =0 \\ \Rightarrow & & m _1 m _2 m _3 m _4 & =\frac{1}{1}=1 \end{array} $$
