Question

If the circle x²+y²=a² intersects the hyperbola xy=c², at 4 pts. P(x₁,y₁), Q(x₂,y₂), R(x₃,y₃) , S(x₄,y₄) then a)x1+x2+x3+x4=0 a)x1+x2+x3+x4=0 b)x1x2x3x4=0 a)y1+y2+y3+y4=0 a)y1y2y3 y₄=0

Answer 1

Dear student
There is a mistake in your question it should have been If the circle x²+y²=a² intersects the hyperbola xy=c², at 4 pts. P(x₁,y₁), Q(x₂,y₂), R(x₃,y₃) , S(x₄,y₄) then which of the following need not hold
a)x1+x2+x3+x4=0
b)x1x2x3x4=y1y2y3y₄=c^4
c)y1+y2+y3+y4=0
d)x₁+y₂+x₃+y₄=0
$$\begin{gathered} \mathrm{The}\mathrm{abscissas}\mathrm{of}\mathrm{the}\mathrm{points}\mathrm{of}\mathrm{the}\mathrm{intersection}\mathrm{of}\mathrm{the}\mathrm{given}\mathrm{curves}\mathrm{are}\mathrm{connected}\mathrm{by} \\ {\mathrm{x}}^2+\frac{{\mathrm{c}}^4}{{\mathrm{x}}^2}={\mathrm{a}}^2\Rightarrow {\mathrm{x}}^4-{\mathrm{a}}^2{\mathrm{x}}^2+{\mathrm{c}}^4=0 \\ \mathrm{As}{\mathrm{x}}_1,{\mathrm{x}}_2,{\mathrm{x}}_3,{\mathrm{x}}_4\mathrm{are}\mathrm{the}\mathrm{roots}\mathrm{of}\mathrm{the}\mathrm{equation} \\ {\mathrm{x}}_1+{\mathrm{x}}_2+{\mathrm{x}}_3+{\mathrm{x}}_4=0,{\mathrm{x}}_1{\mathrm{x}}_2{\mathrm{x}}_3{\mathrm{x}}_4={\mathrm{c}}^4 \\ \mathrm{Similarly}{\mathrm{y}}_1+{\mathrm{y}}_2+{\mathrm{y}}_3+{\mathrm{y}}_4=0,{\mathrm{y}}_1{\mathrm{y}}_2{\mathrm{y}}_3{\mathrm{y}}_4={\mathrm{c}}^4 \end{gathered}$$

Regards