How many circles can be drawn through two given points?

Question

How many cirxles can be drawn through 3 non collinear points?

Answer 1

It takes 3 non collinear points to define one specific circle. With only two points an infinite number of circles can be drawn.

Proof:

Given two points A, B draw the line between them. Then find the perpendicular bisector of the line AB. Any point on the perpendicular bisector is equidistant from the two original points, A and B. A circle with center C and radius AC will then pass through points A and B. There are infinite point C's on the perpendicular bisector so there are infinite circles.

Given three points A, B and D you can find the perpendicular bisector for line segements AB and then the perpendicular bisector fof line segment BC. The two perpedicular bisectors will not be parallel because the points A, B and D are non collinear. This means the two perpeniducar bisectors will intercept at only one point C(like any two intercepting lines). This point C is equidistant from points A, B, and D. A circle with center C and radius AC will then pass through all three of the points. Since there is only one point C that lies on both perpendicular bisectors, there is only one circle possible.

Answer 2

2

Answer 3

infinite no of circles can b drawn from 2 points. only 1 circle can be drawn from 3 non collinear points.

Answer 4

Through two given points infinity circles can be drawn.
Through three non collinear points, one circle.

Answer 5

Through two points, infinitely many. (Imagine the two points as endpoints of a diameter, then "push" the circle down and increase its size so that it still contains the two points.)

The third (noncollinear) point determines the circle uniquely, so just one.

FYI, here's why. The arbitrary equation for a circle is Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 which has six parameters. Each point has two coordinates (x and y). Two points have four coordinates total, less than six, which leaves infinitely many ways to complete the circle equation. Three points have six coordinates total -- if they work (that's the noncollinear requirement), then they specify all six parameters of the equation.

Answer 6

u many as many distinct circles as u want through 2 pts
n only 1 circle can be drawn thru 3 pts

Answer 7

infinite number of circles can be drawn in case of two points but the
answer is one in case of 3 points

Answer 8

Only one by taking the distance b/w two pts. as diameter.