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 \sqrt{2}‘s?)

You also need to know the basic circle equation, (x-h)^2+(y-k)^2=r^2, where (h, k) is the circle’s center and r 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 h and k. Like so:


So, which of the coordinates in the answer choices will result in a hypotenuse of 10? Only (5\sqrt{2},5\sqrt{2}), which will have a hypotenuse of \left(5\sqrt{2}\right)\sqrt{2}, which of course simplifies to 10. 

Does that help?  

Comments (3)

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

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