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 - x_o)² + (y - y_o)² = r², in which x_o is the x-coordinate of the center of the circle, y_o is the y-coordinate of the center of the circle, r is the radius of the circle. Now, we will find x_o, y_o 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 (x_o, y_o).

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

x_o = GC = GD - CD = r - 2

y_o = 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

x_o = 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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