Not sure.....
(1) if we take u = (1,0,0) , v = (1/2 , 1, 0 ) and w = (1/2 ,1/2 ,1/2). the relation is not true
(2) Ratio as 1:1:1
(3) d = 1
Not sure.....
(1) if we take u = (1,0,0) , v = (1/2 , 1, 0 ) and w = (1/2 ,1/2 ,1/2). the relation is not true
(2) Ratio as 1:1:1
(3) d = 1
2) consider the figure,
in triangle BOC,
OBC = 90-c , OCB = 90-B
=> BOC = B+c
by using sine rule,
R_{Î”BOc} = BC2sin(BOC) = a2sin(B+c) = circumradius of Î”ABC
similarly others also = R (of Î”ABC)
.:. ratio = 1:1:1
3)we can notice that x=y will be the only solution for both hyperbola and circle when it is of the form xy=k and too the circle will just fit in between the 2 branches of the hyperbola
(can be understood from a diagram)
so the hyperbola is xy=1 {as b=0 and given that c=a}
so now on solving we get d=2
if it were any other hyperbola,
case-1
for both points of intersection to lie on a straight line (y=x in this case)
only other possibility would be that the points of intersection lie on x-axis or y-axis. so this is not considered.
case-2
in any other case there will be 4 solutions will can't lie on straight line
Thanks Rishabh for Nice explanation
I also confused in that question.