First slide

Definition of a circle

Question

If $(m;, 1 / m;), i = 1, 2, 3, 4$ are concyclic points, then the value of $m_{1} m_{2} m_{3} m_{4},$ is

Moderate

A

1
B

$- 1$
C

0
D

none of these

Solution

Let the equation of the circle be
$x^{2} + y^{2} + 2 g x + 2 y y + c = 0 .$
If $(m, 1 I m)$ Lies on this circle, then
$\begin{array}{r} m^{2} + \frac{1}{m^{2}} + 2 g m + 2 f \frac{1}{m} + c = 0 \\ \Rightarrow m^{4} + 2 g m^{3} + 2 f m + c m^{2} + 1 = 0 \end{array}$
This is a fourth degree equation in $m$ having $m_{1}, m_{2}, m_{3}, m_{4}$ as its roots.
$∴$ Product of the roots $= 1 \Rightarrow m_{1} m_{2} m_{3} m_{4} = \frac{1}{1} = 1$

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