Question

If the circle $x^2+y^2=a^2$ intersects the hyperbola $xy=c^2$ in four points $P(x_1,y_1),Q(x_2,y_2),R(x_3,y_3),S(x_4,y_4)$, then which of the following need not hold

• A.
  
• B.
  
• C.
  
• D.
  

Answer 1

The correct option is D


The abscissae of the points of intersection of the given curves are connected by
$x^2+\frac{c^4}{x^2}=a^2\Rightarrow x^4-a^2{x}^2+c^4=0$
As $x_1,x_2,x_3,x_4$ are the roots of this equation $x_1+x_2+x_3+x_4$ $=0,x_1{x}_2{x}_3{x}_4=c^4$
Similarly $y_1+y_2+y_3+y_4=0,y_1{y}_2{y}_3{y}_4=c^4$