SATHEE: Hyperbola 3 Question 2

Hyperbola 3 Question 2

2.

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

$(1998, 2 M)$ (a) $x_{1} + x_{2} + x_{3} + x_{4} = 0$

(b) $y_{1} + y_{2} + y_{3} + y_{4} = 0$

(d) $y_{1} y_{2} y_{3} y_{4} = c^{4}$

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Answer:

Correct Answer: 2. $(a, b, c, d)$

Solution:

It is given that,

$x^{2} + y^{2} = a^{2}$

and $x y = c^{2}$

We obtain $x^{2} + c^{4} / x^{2} = a^{2}$

$\Rightarrow x^{4} - a^{2} x^{2} + c^{4} = 0$

Now, $x_{1}, x_{2}, x_{3}, x_{4}$ will be roots of Eq. (iii).

Therefore, $Σ x_{1} = x_{1} + x_{2} + x_{3} + x_{4} = 0$ and product of the roots $x_{1} x_{2} x_{3} x_{4} = c^{4}$

Similarly, $y_{1} + y_{2} + y_{3} + y_{4} = 0$

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

Hence, all options are correct.

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Hyperbola 3 Question 2

2.

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