In the coordinate system, points and all lie on the same circle.
To 1 decimal place, how far is the center of that circle from the origin?
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
The circumcentre X of the triangle ABC, whose vertices have position vectors a , b , c respectively, has the following properties:
With the points A ( − 1 , 7 , 5 ) , B ( 1 3 , 9 , 2 ) , C ( 4 , − 8 , 1 ) , the intersection of these two planes is the line ℓ with equation x = 4 4 0 2 6 9 7 + 1 0 6 u y = 4 4 0 8 1 1 − 8 2 u z = u Moreover A X must lie in the plane A B C , and hence must be perpendicular to the vector n = ( b − a ) ∧ ( c − a ) = ⎝ ⎛ − 5 3 4 1 − 2 2 0 ⎠ ⎞ (note that the line ℓ is in the direction of the vector n ). The point on ℓ with this final property is found when u = 1 0 5 7 8 2 2 4 7 1 , and hence X has coordinates ( 5 2 8 9 0 3 5 1 2 5 9 , 1 2 9 0 1 8 6 7 , 1 0 5 7 8 2 2 4 7 1 ) which makes the answer 7 . 1 2 1 4 .