First slide

Hyperbola in conic sections

Question

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

Difficult

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

$y_{1} y_{2} y_{3} y_{4} = c^{4}$

Solution

$Solving x y = c^{2} and x^{2} + y^{2} = a^{2}, we have$
$x^{2} + \frac{c^{4}}{x^{2}} = a^{2}$
$\begin{array}{lrl} \Rightarrow x^{4} - a^{2} x^{2} + c^{4} = 0 \\ \Rightarrow Σ x_{i} = 0 and x_{1} x_{2} x_{3} x_{4} = c^{4} \end{array}$
$Similarly, if we eliminate y, then Σ y_{i} = 0 and y_{1} y_{2} y_{3} y_{4}$
$= c^{4}$

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