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)

$t_{1} t_{2} t_3 t_{4} = 1$

Answer 2

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, $t_{1}+ t_{2}+ t_{3} + t_{4} = \frac{2gc}{c^{2}} = -\frac{2g}{c}$