HOW TO FIND THE CENTER AND RADIUS OF A CIRCLE FROM ITS EQUATION

Find the center and radius of the following circles.

Example 1 :

x² + y² = 25

Solution :

The given equation of the circle is in the form of

x² + y² = r²

So, the center of the given circle is (0, 0).

Radius :.

r² = 25

r = 5 units

Example 2 :

(x - 1)² + (y - 3)² = 9

Solution :

The given equation of the circle is in the form of

(x - h)² + (y - k)² = r²

Center :

(h, k) = (0, 0)

Radius :.

r² = 9

r = 3 units

Example 3 :

(x + 3)² + (y - 5)² = 15

Solution :

The given equation of the circle is in the form of

(x - h)² + (y - k)² = r²

Center :

(h, k) = (-3, 5)

Radius :.

r² = 15

r = √15 units

Example 4 :

x² + y²  - 4x  - 6y + 9 = 0

Solution :

The given equation of circle is in general form.

Comparing 

x² + y² - 4x - 6y + 9 = 0

and 

x² + y² + 2gx + 2fy + c = 0

we get

2g = -4 ----> g = -2 ----> -g = 2

2f = -6 ----> f = -3 ----> -f = 3

Center :

(-g, -f) = (2, 3)

Radius :

r = √(g² + f² - c)

= √(4 + 9 - 9)

= √4

= 2 units

Example 5 :

Find the center of the circle described on the line joining the points (1, 2) and (2, 4) as its diameter.

Solution :

Center of the circle = Midpoint of the diameter

Substitute (x₁, y₁) = (1, 2) and (x₂, y₂) = (2, 4).

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