Question

If the circle $x^2+y^2=a^2$ intersects the hyperbola $xy=c^2$ in four points $P(x_1,y_1),Q(x_2,y_2),R(x_3,y_3),S(x_4,y_4)$, then -

• A. $X_1+X_2+X_3+X_4=0$
• B. $Y_1+Y_2+Y_3+Y_4=0$
• C. $X_1{X}_2{X}_3{X}_4=c^4$
• D. $Y_1{Y}_2{Y}_3{Y}_4=c^4$

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A $X_1+X_2+X_3+X_4=0$
B $Y_1+Y_2+Y_3+Y_4=0$
C $Y_1{Y}_2{Y}_3{Y}_4=c^4$
D $X_1{X}_2{X}_3{X}_4=c^4$

Since, the circle $x^2+y^2=a^2$ intersects the hyperbola $xy=c^2$
Therefore, $x^2+\frac{c^4}{x^2}=a^2$
$\Rightarrow x^4-a^2{x}^2+c^4=0$
now sum of the roots: $x_1+x_2+x_3+x_4=0$
and product of the roots $x_1{x}_2{x}_3{x}_4=c^4$
Repeat the same for $y$