Mathematics - Coordinate Geometry Mixed Question with Solution | TestHub

The circle is $x^2+y^2-2x-4y-20=0$; let the hyperbola be $xy=c^2$ with points $\left(ct,\frac{c}{t}\right)$. Substituting: $c^2{t}^4-2c{t}^3-20t^2-4ct+c^2=0$. With roots $t_1,t_2,t_3,t_4:\sum t_1=\frac{2}{c},\sum t_1{t}_2=-\frac{20}{c^2},\sum t_1{t}_2{t}_3=\frac{4}{c},t_1{t}_2{t}_3{t}_4=1$. $\sum x_1=c\sum t_1=2\to \mathrm{P}$. (D) $\sum x_1{x}_2=c^2\sum t_1{t}_2=-20\to \mathrm{S}$. (B) $\sum x_1^2={\left(\sum x_1\right)}^2-2\sum x_1{x}_2=4+40=44\to \mathrm{R}$. $\sum y_1=c\sum \frac{1}{t_1}=c\cdot \frac{\sum t_1{t}_2{t}_3}{t_1{t}_2{t}_3{t}_4}=4$ and $\sum y_1{y}_2=-20:\sum y_1^2=16+40=56\to \mathrm{Q}$.