If the coordinates of four concyclic points on the rectangular hyperbola xy = c2 are (cti, c / ti ), i = 1, 2, 3, 4 then

Question

If the coordinates of four concyclic points on the rectangular hyperbola xy = c2 are (cti, c / ti ), i = 1, 2, 3, 4 then (A)  t1 t2 t3 t4 = -1  (B)  t1 t2 t3 t4 = 1  (C)  t1 t3 = t2 t4  (D)  t1+ t2+ t3 + t4 = c2

Tardigrade Admin · Accepted Answer

Let the points lie on the circle x2 +y 2+2gx +2fy + 2fy + k = 0, then c2t2i + (c2/t2i) +2gcti + 2f (c/ti) + k = 0 ⇒ c2t4i + 2gct3i + kti3 + 2fcti + c2 = 0 Its roots are t1, t2, t3, t4 so t1 t2 t3 t4 = (c2/c2) = 1 Also, t1+ t2+ t3 + t4 = (2gc/c2) = -(2g/c)

Q. If the coordinates of four concyclic points on the rectangular hyperbola $x y = c^{2}$ are $(c t_{i}, c / t_{i}), i = 1, 2, 3, 4$ then

$t_{1} t_{2} t_{3} t_{4} = - 1$

$t_{1} t_{2} t_{3} t_{4} = 1$

$t_{1} t_{3} = t_{2} t_{4}$

$t_{1} + t_{2} + t_{3} + t_{4} = c^{2}$

Q. If the coordinates of four concyclic points on the rectangular hyperbola xy=c2 are (cti​,c/ti​),i=1,2,3,4 then

t1​t2​t3​t4​=−1

t1​t2​t3​t4​=1

t1​t3​=t2​t4​

t1​+t2​+t3​+t4​=c2

Let the points lie on the circle x2+y2+2gx+2fy+2fy+k=0, then c2ti2​+ti2​c2​+2gcti​+2fti​c​+k=0 ⇒c2ti4​+2gcti3​+kti3​+2fcti​+c2=0 Its roots are t1​,t2​,t3​,t4​ so t1​t2​t3​t4​=c2c2​=1 Also, t1​+t2​+t3​+t4​=c22gc​=−c2g​

Q. If the coordinates of four concyclic points on the rectangular hyperbola $x y = c^{2}$ are $(c t_{i}, c / t_{i}), i = 1, 2, 3, 4$ then

$t_{1} t_{2} t_{3} t_{4} = - 1$

$t_{1} t_{2} t_{3} t_{4} = 1$

$t_{1} t_{3} = t_{2} t_{4}$

$t_{1} + t_{2} + t_{3} + t_{4} = c^{2}$