If $\left(\mathrm{m}_{\mathrm{i}}, \frac{1}{\mathrm{~m}_{\mathrm{i}}}\right), \mathrm{m}_{\mathrm{i}}\gt0,…
If $\left(\mathrm{m}_{\mathrm{i}}, \frac{1}{\mathrm{~m}_{\mathrm{i}}}\right), \mathrm{m}_{\mathrm{i}}\gt0, \mathrm{i}=1,2,3,4$ are four distinct points on a circle, then the product $\mathrm{m}_1 \mathrm{~m}_2 \mathrm{~m}_3 \mathrm{~m}_4$ is equal to
-1
1
0
2
Solution
Let $\left(m_i, \frac{1}{m_i}\right)$ lie on the circle
$x^2+y^2+2 \mathrm{~g} x+2 \mathrm{f} y+\mathrm{c}=0$
where $\mathrm{i}=1,2,3,4$
$\begin{aligned}
& \therefore \quad\left(m_i\right)^2+\left(\frac{1}{m_i}\right)^2+2 \mathrm{gm}_{\mathrm{i}}+\frac{2 \mathrm{f}}{m_i}+\mathrm{c}=0 \\
& \quad \Rightarrow \mathrm{~m}_{\mathrm{i}}^2+\frac{1}{\mathrm{~m}_{\mathrm{i}}^2}+2 \mathrm{gm}_{\mathrm{i}}+\frac{2 \mathrm{f}}{\mathrm{~m}_{\mathrm{i}}}+\mathrm{c}=0 \\
& \quad \Rightarrow \mathrm{~m}_{\mathrm{i}}^4+1+2 \mathrm{gm}_{\mathrm{i}}^3+2 \mathrm{fm}_{\mathrm{i}}+\mathrm{cm}_{\mathrm{i}}^2+2=0 \\
& \quad \Rightarrow \mathrm{~m}_{\mathrm{i}}^4+2 \mathrm{gm}_{\mathrm{i}}^3+\mathrm{cm}_{\mathrm{i}}^2+2 \mathrm{fm}_{\mathrm{i}}+1=0
\end{aligned}$
$\mathrm{m}_1, \mathrm{~m}_2, \mathrm{~m}_3, \mathrm{~m}_4$ are roots of the above equation
$\therefore \quad$ product of roots $=\frac{\mathrm{e}}{\mathrm{a}}=\frac{1}{1}=1$
$\therefore \quad \mathrm{m}_1 \mathrm{~m}_2 \mathrm{~m}_3 \mathrm{~m}_4=1$