Hi @gregmat

In above question some information seems to be missing. How can we be sure about the centres of the circles if it is not given that x=0 or y=2 or x=2 or y=0 are tangents to circles? We cannot just realise on it visually right? Because visually y = 0.000001 x would also seem to be a tangent but would change the answer. No?

On the GRE, if the diagram shows that points or lines are intersecting, then they donâ€™t have to be labeled

@gregmat Yesâ€¦ But only information we can deduce for sure from above diagram is that both circles are passing through (0,0) and (2,2)â€¦ And that would not be sufficient to find centres of the circles. We can only say that centres would be lying somewhere on the line x+y=2 (perpendicular to the cord joining these two points). How can we be sure about centres from the given information?

Good question. If a circle can pass through both of those points and have a different radius, then the question is bad because of its ambiguity. Can you provide a diagram of a circle with a different radius passing through those two points?

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Hi @gregmat ,

Yes, sureâ€¦

In above diagram, Circle is centred at (2.1, -0.1) with radius squareRoot(4.42)â€¦ And It is also passing through both of those points as can be seen in above diagram. We can create infinite such circles passing through these two points by adjusting radius and centre accordingly.

Note: You may further verify same by putting both points in equation of circle.

(x-2.1)^2 + (y+0.1)^2 = 4.42

In that diagram it doesnâ€™t look like the origin (0,0) and the point (2,2) are being intersected

Hi @gregmat ,

It is being intersected actually. You may notice that wolfram alpha has moved (0,0) little bit upward and (2,2) little bit downward to make things clear. One way is to count dashes on axis, or in other way you can also put points in equation of circle and verify same too.

I also drew it again with increased axis ranges, You may verify same now.

Yes, indeed. Good call. Weâ€™ll clarify the question. @Leaderboard

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Iâ€™ve added the â€śquarter-circleâ€ť wording, which should fix the flaw (kautsgreâ€™s suggested equation of circle would not be a quarter-circle in this context). The intersection points have also been mentioned explicitly.

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