Question

If $(mᵢ, 1/mᵢ)$ are four distinct points on a circle, then

• A. $m_{1}m_{2}m_{3}m_{4}=1$
• B. $m_{1}m_{2}m_{3}m_{4}=-1$
• C. $m_{1}m_{2}m_{3}m_{4}=1/2$
• D. $1/m_{1}+1/m_{2}+1/m_{3}+1/m_{4}=1/4$

Hint:

If $(mi, 1/mi)$ are four distinct points on a circle, then

Answer 1

The Correct Option is A

Step 1: Concept
Substitute the coordinates $(m, 1/m)$ into the general equation of a circle $x^2 + y^2 + 2gx + 2fy + c = 0$.
Step 2: Analysis
$m^2 + (1/m)^2 + 2gm + 2f/m + c = 0 \Rightarrow m^4 + 2gm^3 + cm^2 + 2fm + 1 = 0$.
Step 3: Evaluation
The values $m_1, m_2, m_3, m_4$ are the roots of this fourth-degree equation.
Step 4: Conclusion
The product of the roots is the constant term divided by the coefficient of $m^4$, which is $1/1 = 1$.
Final Answer: (a)