Question

If the coordinates of four concyclic points on the rectangular hyperbola $xy = c^2$ are $(ct_i, c / t_i ), \,i = 1, 2, 3, 4$ then

• A. $ t_{1} t_{2} t_3 t_{4} = -1 $
• B. $ t_{1} t_{2} t_3 t_{4} = 1 $
• C. $ t_{1} t_{3} = t_{2} t_{4} $
• D. $ t_{1}+ t_{2}+ t_3 + t_{4} = c^{2} $

Answer 1

Answer: (B)

Let the points lie on the circle
$x^{2} +y ^{2}+2gx +2fy + 2fy + k = 0$, then
$c^{2}t^{2}_{i} + \frac{c^{2}}{t^{2}_{i}} +2gct_{i} + 2f \frac{c}{t_{i}} + k = 0$
$\Rightarrow \quad c^{2}t^{4}_{i} + 2gct^{3}_{i} + kt_{i}^{3} + 2fct_{i} + c^{2} = 0$
Its roots are $t_{1}, t_{2}, t_3, t_{4}$ so
$t_{1} t_{2} t_3 t_{4} = \frac{c^{2}}{c^{2}} = 1$
Also, $t_{1}+ t_{2}+ t_{3} + t_{4} = \frac{2gc}{c^{2}} = -\frac{2g}{c}$