How to Find the Center and the Radius of Circles

To find the center and radius of a circle from its equation, you need to find the equation of the circle in is the centerradius form.

Rules for Finding the Center and the Radius of Circles

To find the center and the radius of a circle using the equation of the circle:

• Write the equation of the circle in standard form: $$(x- h)^2+( y-k)^2= r^2$$,
• The center of the circle is at (h,k), and its radius is r.

Examples

Find the Center and the Radius of Circles – Example 1:

Identify the center and the radius of each circle.

$$x^2+ y^2-4x+3=0$$

Solution:

$$(x- h)^2+( y-k)^2= r^2$$ is the circle equation with a radius r, centered at (h,k).
Rewrite $$x^2+ y^2-4x+3=0$$ in the standard form:
$$x^2+ y^2-4x+3=0→(x-2)^2+y^2=1^2$$
Then, the center is at: (2,0) and $$r=1$$

Find the Center and the Radius of Circles – Example 2:

Identify the center and the radius of each circle.

$$8x+x^2+10y=8- y^2$$

Solution:

Rewrite the equation in standard form:
$$8x+x^2+10y=8- y^2→(x-(-4))^2+(y-(-5))^2=7^2$$
Then, the center is at $$(-4,-5)$$ and the radius is 7.

Find the Center and the Radius of Circles – Example 3:

$$8x+x^2-2y=8- y^2$$

Solution:

$$(x- h)^2+( y-k)^2= r^2$$ is the circle equation with a radius r, centered at (h,k).
Rewrite $$8x+x^2-2y=8- y^2$$ in the standard form:
$$(x-(-4))^2+(y-1)^2=5^2$$
Then, the center is at $$(-4,1)$$ and $$r=5$$

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