Neka kružnica polumjera
![r](/media/m/3/d/f/3df7cc5bbfb7b3948b16db0d40571068.png)
siječe hiperbolu
![xy=1](/media/m/b/f/3/bf324011385553a132a09b135fd0ebd4.png)
u četiri točke
![P_1(x_1,y_1)](/media/m/9/0/1/901ca0f9d9a400d63aa9d4b2debb337d.png)
,break
![P_2(x_2,y_2)](/media/m/d/6/a/d6a227214552edd4a1c2d81b20aec7d2.png)
,
![P_3(x_3,y_3)](/media/m/f/4/a/f4a5591690fb430e087b446875c8c3f4.png)
,
![P_4(x_4,y_4)](/media/m/3/e/7/3e725e7117366a3cd878e6eff949f37d.png)
. Dokaži da vrijedi:
![x_1x_2x_3x_4=y_1y_2y_3y_4=1](/media/m/9/f/f/9ff47965fa8de6428b243e4979654e18.png)
,
![{\sum _{k=1}^4|OP_k|^2=4r^2}](/media/m/a/c/b/acb40cd7282512f06b7e92656539f97e.png)
(
![O](/media/m/9/6/0/9601b72f603fa5d15addab9937462949.png)
je ishodište koordinatnog sustava).
%V0
Neka kružnica polumjera $r$ siječe hiperbolu $xy=1$ u četiri točke $P_1(x_1,y_1)$,\break $P_2(x_2,y_2)$, $P_3(x_3,y_3)$, $P_4(x_4,y_4)$. Dokaži da vrijedi:
$a)$ $x_1x_2x_3x_4=y_1y_2y_3y_4=1$,
$b)$ ${\sum _{k=1}^4|OP_k|^2=4r^2}$ ($O$ je ishodište koordinatnog sustava).