which statement is true and why?

Question

Let A and B be two distinct points in the plane, d their distance apart, and r a given positive integer. Then

(A) there always exists a circle of radius r passing through A and B
(B) if d ≤ 2r then there exists a unique circle of radius r passing through A and B
(C) there exists a circle of radius r passing through A and B only if d ≥ 2r
(D) if d < 2r then there exist two circles of radius r passing through A and B
(E) there exists a circle of radius r passing through A and B only if d = 2r

I didn't understand this question. Could anyone help pls?

Answer 1

Do These Circles Help?

From the description given, I'm assuming $d$ and $r$ must be chosen once and not changed for the duration of an example, so you could not use an example with $2$ different values for $d$ or multiple values for $r$.

If $d > 2r$, then either A or B can be on the circle, but not both.

If $d == 2r$, then $d$ is the diameter and A and B are both on the circle.

if $d < 2r$, then you can move the circle around until both A and B are on the circle... and you can also place a second copy of the circle (with the same radius r) in an orientation so that A and B pass through both circles.

I would choose answer $d$.

Answer 2

AB is supposed to be chord of a circle. Length of chord (d) can be maximum the length of diameter of the cirlcle (2r). So $d\le 2r$. The before mentioned condition satisfied, then you can draw two circles passing through A and B. The center of the circles will be on each side of the chord. So two circles can be drawn, When you increase the length of the chord d (keeping r constant) the distance between the center of the circles will reduce. When $d = 2r$, the distance will be zero, so there will be only one circle and the center will lie on the chord.