Circle equation.If given 2 points decide what is the equation of the circle that passing through the points.

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

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It is not possible to obtain the equation of a circle that passes through 2 points because there can be an unlimited number of circles that can pass through the two points.

Like we need two points to define a line, 1 point is not enough; for a circle 3 points are essential, 2 are not enough.

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

The best way to realise the problem is to take a numeric example.

We'll choose 2 points (3,1) and (-1,3).

The center of the circle is on the line 3x-y-2=0

We'll write the equation of the circle:

(x - h)^2 + (y - k)^2 = r^2

The point (3,1) is on the circle if and only if:

(3 - h)^2 + (1 - k)^2 = r^2 (1)

The point (-1,3) is on the circle if and only if:

(-1 - h)^2 + (3 - k)^2 = r^2 (2)

The center is located on the line 3x-y-2=0, if and only if:

3h - k = 2 (3)

We'll put (1) = (2):

(3 - h)^2 + (1 - k)^2 = (-1 - h)^2 + (3 - k)^2

We'll expand the squares:

9 - 6h + h^2 + 1  -2k + k^2 = 1 + 2h + h^2 + 9 - 6k + k^2

We'll eliminate like terms;

- 6h - 2k = 2h - 6k

We'll move all terms to one side:

- 6h - 2k - 2h + 6k = 0

We'll combine like terms:

-8h + 4k = 0

We'll divide by 4:

k = 2h (4)

We'll substitute (4) in (3):

3h - 2h = 2

h = 2

k = 4

To determine the radius, we'll substitute h and k in (1):

(3 - 2)^2 + (1 - 4)^2 = r^2

1 + 9 = r^2

r^2 = 10

The equation of the circle is:

(x - 2)^2 + (y - 4)^2 = 10

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