Question

If $(x_i,\frac{1}{x_i})$ where $i=1,2,3,4$ are four distinct points on a circle, then $x_1\cdot x_2\cdot x_3\cdot x_4$ is equal to

• A. $4$
• B. $-1$
• C. $1$
• D. Zero

Answer 1

Answer: (C)

$1$

Let equation of the circle be,

$x^2+y^2=r^2$ .... $\left(i\right)$

For points, $(x,\frac{1}{x})$ the equation of circle becomes

$x^2+\frac{1}{x^2}=r^2$

$⟹x^4+1=r^2{x}^2$

$⟹x^4-r^2{x}^2+1=0$ .... $\left(ii\right)$

$(x_i,\frac{1}{x_i});i=1,2,3,4$ are points on the circle

So, $x_1,x_2,x_3,x_4$ becomes roots of $\left(ii\right)$

Thus, $x_1.x_2.x_3.x_4=\frac{1}{1}=1$ .... [Using relation between coefficients and roots]

Hence, option C is correct.