# Equation of the circle.Write the equation of circle that passes through the point (1,4) and the center of circle is (2,1).

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The circle passes through the point (1, 4) and the center is (2, 1). The radius of the circle is the distance between the two points.

r = sqrt [(2 - 1)^2 + ( 4 - 1)^2]

=> sqrt [ 1 + 9]

=> sqrt 10

The equation of the circle is (x - 2)^2 + (y - 1)^2 = [sqrt 10]^2

=> x^2 - 4x + 4 + y^2 - 2y + 1 = 10

=> x^2 + y^2 - 4x - 2y - 5 = 0

**The required circle is x^2 + y^2 - 4x - 2y - 5 = 0**

We'll write the equation of the circle:

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

The center of the circle has the coordinates C(h ; k).

We know, from enunciation, that h = 2 and k = 1.

We'll substitute them into the equation:

(x - 2)^2 + (y - 1)^2 = r^2

We'll determine the radius considering the constraint from enunciation that the circle is passing through the point (1,4).

If the circle is passing through the point (1,4), then the coordinates of the point are verifying the equation of the circle:

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

1^2 + 3^2 = r^2

1 + 9 = r^2

10 = r^2

r = sqrt10

The equation of the circle is:

(x - 2)^2 + (y - 1)^2 = 10