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# Find circle equation

If a circle passes the origin A (0, 0) and point B (0, 6). The circle is tangent to the line x = -2. Then what is the equation of the circle?

Solution:

The equation of a circle is: (x - xo)2 + (y - yo)2 = r2, in which xo is the x-coordinate of the center of the circle, yo is the y-coordinate of the center of the circle, r is the radius of the circle. Now, we will find xo, yo and r.

Connect AB, AB is a chord. The chord of a circle has the property: If a line is perpendicular and bisects a chord, then this line must be a diameter of the circle.

Find the midpoint of AB, which is the point C. Draw a line passing the point C and perpendicular to AB, this line intersects the circle at point D and point E. The midpoint of DE is the center of the circle. Name the center of the circle as point G, its coordinate is (xo, yo).

Connect AG, AG is the radius of the circle. Triangle ACG is a right triangle.

xo = GC = GD - CD = r - 2
yo = CA = BA/2 = 6/2 = 3
GA = r
In right triangle ACG,
GC2 + CA2 = GA2
(r - 2)2 + 32 = r2
r2 - 4r + 4 + 9 = r2
4r = 13
r = 13/4
xo = GC = r - 2 = 13/4 - 8/4 = 5/4

Therefore, the equation of the circle is: (x - 5/4)2 + (y - 3)2 = 169/16.

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