Question

If a circle cuts the rectangular hyperbola xy = 1 in the points $(x_r,y_r)$; $r=1,2,3,4$, then $x_1{x}_2{x}_3{x}_4=y_1{y}_2{y}_3{y}_4$is equal to

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

The correct option is A 1

Let the equation of the circle be $x^2+y^2+2gx+2fy+k=0$
The equation of the hyperbola is $xy=c^2$
Eliminating y from these two equations, we get
$x^2+\frac{1}{x^2}+2gx+2f\left(\frac{1}{x}\right)+k=0$
$\Rightarrow$ $x^4+2g{x}^3+k{x}^2+2fx+1=0$
This is a fourth degree equation in x giving four values of x say
$x_1,x_2,x_{3}and{x}_4.$
$\therefore$ $x_1{x}_2{x}_3{x}_4=1$
Corresponding to every value of x, there is a value of y given by xy = 1
$\therefore$ $y_1{y}_2{y}_3{y}_4=\frac{1}{x_1{x}_2{x}_3{x}_4}=1$