• question_answer If a circle cuts a rectangular hyperbola $xy={{c}^{2}}$ in A, B, C, D and the parameters of these four points be ${{t}_{1}},\ {{t}_{2}},\ {{t}_{3}}$ and ${{t}_{4}}$ respectively. Then [Kurukshetra CEE 1998] A)            ${{t}_{1}}{{t}_{2}}={{t}_{3}}{{t}_{4}}$                                         B)            ${{t}_{1}}{{t}_{2}}{{t}_{3}}{{t}_{4}}=1$ C)            ${{t}_{1}}={{t}_{2}}$                D)            ${{t}_{3}}={{t}_{4}}$

Let equation of circle is ${{x}^{2}}+{{y}^{2}}={{a}^{2}}$            Parametric form of $xy={{c}^{2}}$ are $x=ct,\,\,\,y=\frac{c}{t}$            Þ ${{c}^{2}}{{t}^{2}}+\frac{{{c}^{2}}}{{{t}^{2}}}={{a}^{2}}$ Þ ${{c}^{2}}{{t}^{4}}-{{a}^{2}}{{t}^{2}}+{{c}^{2}}=0$                    Product of roots will be, ${{t}_{1}}{{t}_{2}}{{t}_{3}}{{t}_{4}}=\frac{{{c}^{2}}}{{{c}^{2}}}=1$.