Q65 / 120 · VITEEE 2009 · Mathematics
If the circle $x^{2}+y^{2}=a^{2}$ intersects the hyperbola $x y=c^{2}$ in four points $\left(x_{1}, y_{1}\right)$ for $i=1,2,3$ and 4, then $y_{1}+y_{2}+y_{3}+y_{4}$ equals
- Correct answerA0
- B$c$
- C$a$
- D$c^{4}$
Solution
$$
\begin{array}{l}
x^{2} y^{2}=c^{4} \\
\Rightarrow y^{2}\left(a^{2}-y^{2}\right)=c^{4} \\
\Rightarrow y^{4}-a^{2} y^{2}+c^{4}=0
\end{array}
$$
Let $y_{1}, y_{2}, y_{3}$ and $y_{4}$ are the roots.
$$
\therefore y_{1}+y_{2}+y_{3}+y_{4}=0
$$
Hence, the correct answer is option: 'A'.