Hyperbola 3 Question 2
2. If the circle $x^{2}+y^{2}=a^{2}$ intersects the hyperbola $x y=c^{2}$ in four points $P\left(x _1, y _1\right), Q\left(x _2, y _2\right), R\left(x _3, y _3\right), S\left(x _4, y _4\right)$, then
$(1998,2 M)$ (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) $y _1 y _2 y _3 y _4=c^{4}$
Analytical & Descriptive Question
Show Answer
Answer:
Correct Answer: 2. $(a, b, c, d)$
Solution:
- It is given that,
$$ x^{2}+y^{2}=a^{2} $$
and $\quad x y=c^{2}$
We obtain $x^{2}+c^{4} / x^{2}=a^{2}$
$$ \Rightarrow \quad x^{4}-a^{2} x^{2}+c^{4}=0 $$
Now, $x _1, x _2, x _3, x _4$ will be roots of Eq. (iii).
Therefore, $\quad \Sigma x _1=x _1+x _2+x _3+x _4=0$ and product of the roots $x _1 x _2 x _3 x _4=c^{4}$
Similarly, $\quad y _1+y _2+y _3+y _4=0$
and $\quad y _1 y _2 y _3 y _4=c^{4}$
Hence, all options are correct.
