Which of the following equations describes a circle with radius 10 that passes through the origin when graphed in the xy-plane?
A) (x – 5)² + (y+5)² = 10
B) (x – 5)² + (y+5)² = 100
C) (x – 10)² + (y+10)² = 10
D) (x – 5√2)² + (y+5√2)² = 100
Clearly, A) is out because that one does not have a radius of 10. What is the most time-efficient way to solve this? Sketch and eyeball?
The move here is to recognize that it’s a right triangle question. (There’s a subtle clue to this in the answer choices–see those 
‘s?)
You also need to know the basic circle equation, 
, where 
is the circle’s center and 
is the radius, but from your question I sense that you have this bit down.
Recognize that when a circle goes through the origin, its radius will be the hypotenuse of a right triangle with legs 
and 
. Like so:
So, which of the coordinates in the answer choices will result in a hypotenuse of 10? Only 
, which will have a hypotenuse of 
, which of course simplifies to 10.
Does that help?
Comments (3)
Thank you, Mike! (The image, btw, doesn’t appear on my browsers, but very helpful explanation.)
Just as an alternate method, after canceling choices A & C (because they do not have radii of 10), I plugged in (0,0) because the question stated that the circle passed through the origin. Only answer choice D yields a true statement.
I know the answer had to have something bigger then 10 for the r^2 so b and d are left but something fishy about b that would make the r =50 (2 x 5^2) so when you square( 5sq root 2) you get 50 and twice that leaves only d