If the circle x2 + y2 = a2 intersects the hyperbola xy = c2 in four points P(x1, y1) Q(x2, y2). R(x3, y3) and S(x4, y4) then

Question

If the circle x2 + y2 = a2 intersects the hyperbola xy = c2 in four points P(x1, y1) Q(x2, y2). R(x3, y3) and S(x4, y4) then (A) x1+x2+x3 + x4 = 0 (B) y1 + y2 + y3 + y4 = 2 (C) x1 x2 x3 x4 = 2 c4 (D) y1 y2 y3 y4 = 2 c 4

Tardigrade Admin · Accepted Answer

Given, x2+y2=a2 and x y=c2 ∴ x2+((c2/x))2=a2 ⇒ x4-a2 x2+c4=0 ∴ x1+x2+x3+x4=0

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

$x_{1} + x_{2} + x_{3} + x_{4} = 0$

$y_{1} + y_{2} + y_{3} + y_{4} = 2$

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

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

Given, $x^{2} + y^{2} = a^{2}$ and $x y = c^{2}$
$∴ x^{2} + (\frac{c ^{2}}{x})^{2} = a^{2}$
$\Rightarrow x^{4} - a^{2} x^{2} + c^{4} = 0$
$∴ x_{1} + x_{2} + x_{3} + x_{4} = 0$

Q. If the circle x2+y2=a2 intersects the hyperbola xy=c2 in four points P(x1​,y1​)Q(x2​,y2​).R(x3​,y3​) and S(x4​,y4​) then

x1​+x2​+x3​+x4​=0

y1​+y2​+y3​+y4​=2

x1​x2​x3​x4​=2c4

y1​y2​y3​y4​=2c4