Question

If $(a,\frac{1}{a}),(b,\frac{1}{b}),(c,\frac{1}{c})$ & $(d,\frac{1}{d})$ are four distinct points on a circle of radius $4$ units, then $abcd$ is equal to?

• A. $4$
• B. $16$
• C. $1$
• D. $Noneofthese$

Answer 1

The correct option is C $1$

Let the equation of circle be $x^2+y^2=4^2{x}^2+y^2=16$

$(a,\frac{1}{a})$ passes through circle so $a^2+\frac{1}{a^2}=16a^4-16a^2+1=0$

$(b,\frac{1}{b})$ passes through circle so $b^2+\frac{1}{b^2}=16b^4-16b^2+1=0$

$(c,\frac{1}{c})$ passes through circle so $c^2+\frac{1}{c^2}=16c^4-16c^2+1=0$

$(d,\frac{1}{d})$ passes through circle so $d^2+\frac{1}{d^2}=16d^4-16d^2+1=0$

So generally , $x^4-x^2+1=0$ and $a,b,c,d$ are roots of the equation.

The product of roots $abcd$ is given as $1$