CircleHard
Question
If the circle x2 + y2 = a2 intersects the hyperbola xy = c2 in four points P(x1, y1), Q(x2, y2), R(x3, y3), S(x4, y4), then
Options
A.x1 + x2 + x3 + x4 =0
B.y1 + y2 + y3 + y4 = 0
C.x1x2x3x4 = C4
D.y1y2y3y4 = C4
Solution
It is given that
x2 + y2 = a2 .....(i)
and xy = c2 .....(ii)
We obtain x2 + c2 / x2 = a2
⇒ x4 - a2x2 + c4 = 0
Now, x1, x2, x3, x4, will be roots of Eq. (iii)
Therefore, ∑ x1 = x2 + x3 + x3 + x4 = 0
and product of the roots
x1x2x3x4 = c4
Similarl y, y1 + y2 + y3 + y4 = 0 and y1y2y3y4 = c4
Hence, all options are correct.
x2 + y2 = a2 .....(i)
and xy = c2 .....(ii)
We obtain x2 + c2 / x2 = a2
⇒ x4 - a2x2 + c4 = 0
Now, x1, x2, x3, x4, will be roots of Eq. (iii)
Therefore, ∑ x1 = x2 + x3 + x3 + x4 = 0
and product of the roots
x1x2x3x4 = c4
Similarl y, y1 + y2 + y3 + y4 = 0 and y1y2y3y4 = c4
Hence, all options are correct.
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