Question

If \( \left(x_1, \frac{1}{x_1}\right), \left(x_2, \frac{1}{x_2}\right), \left(x_3, \frac{1}{x_3}\right), \left(x_4, \frac{1}{x_4}\right) \) lie on the boundary of a circle of radius 4, then the value of \( x_1 x_2 x_3 x_4 \) is:

• A. \( 1 \)
• B. \( 2 \)
• C. \( 4 \)
• D. \( \frac{1}{4} \)

Hint:

Points on Circle and Reciprocal Coordinates}
If \( (x, \frac{1}{x}) \) lies on a circle, product invariants often help
For symmetrical identities involving multiplicative inverses, \( x_i \cdot \frac{1}{x_i} = 1 \)
This trick is common in reciprocal coordinate pair problems

Answer 1

The Correct Option is A

The circle's equation is symmetric in the form: Let \( P_i = \left(x_i, \frac{1}{x_i}\right) \) Use the concept of radical circle property: If all points \( P_i \) lie on the circle, then a known trick for such symmetric reciprocal pairs: \[ x_i \cdot \frac{1}{x_i} = 1 \Rightarrow x_1 x_2 x_3 x_4 \cdot \frac{1}{x_1 x_2 x_3 x_4} = 1 \Rightarrow x_1 x_2 x_3 x_4 = 1 \] This is a known result from transformation geometry for such reciprocal pairings.