If the circle x^2+y^2=a^2 intersects the hyperbola x y=c^2...

If the circle $x^{2}+y^{2}=a^{2}$ intersects the hyperbola $x y=c^{2}$ in four points $\mathrm{P}\left(x_{1}, y_{1}\right), Q\left(x_{2}, y_{2}\right), R\left(x_{3}, y_{3}\right), S\left(x_{4}, y_{4}\right)$, 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) $y_{1} y_{2} y_{3} y_{4}=c^{4}$

If the circle $x^{2}+y^{2}=a^{2}$ intersects the hyperbola $x y=c^{2}$ in four points $\mathrm{P}\left(x_{1}, y_{1}\right), Q\left(x_{2}, y_{2}\right), R\left(x_{3}, y_{3}\right), S\left(x_{4}, y_{4}\right)$, 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) $y_{1} y_{2} y_{3} y_{4}=c^{4}$

Key Concepts

Vieta's Formulas

Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots. When the elimination process applied to the system of equations yields a polynomial whose roots are the x-coordinates (or y-coordinates) of the intersection points, Vieta's formulas can be used to directly determine sums and products of those roots. This method bridges the gap between algebraic manipulation and geometric properties of the curves involved.

Symmetry in Geometric Equations

Many geometric figures have inherent symmetries, which often simplify the analysis of their intersections. For example, a circle is symmetric about both axes and the origin. When intersecting with another symmetric curve or one that exhibits specific invariant properties, the sum of the coordinates or other symmetric functions of the intersection points can yield simplified relationships. This symmetry is key in deducing properties like the total sum or product of the coordinates.

Intersection of Curves

This concept deals with finding the common solutions to two or more equations representing curves in the plane. When two curves, such as a circle and a hyperbola, intersect, the coordinates of their intersection points satisfy both equations. The methods used include substitution and elimination, which transform the system of equations into a single equation whose solutions correspond to the coordinates of the intersection points.

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