Equation of a circle?

Question

Hi.
How can I find out the equation of a circle? What is given is that it has a center at (4,1) and a radius of 3. Could you please explain step by step so I can learn from this?
Thank you.

Answer 1

Definition: A circle is the set of points equidistant from a point C(h,k) called the center. The fixed distance r from the center to any point on the circle is called the radius.

The standard equation of a circle with center C(h,k) and radius r is as follows:

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


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Example 1: Find the equation of a circle whose center is at (2, - 4) and radius 5.

Solution to Example 1:

given (h , k ) = (2 , - 4) and r = 5

substitute h, k and r in the standard equation

(x - 2)^2 + (y - (- 4))^2 = 25

(x - 2)^2 + (y + 4)^2 = 25

Answer 2

If you are spatially inclined and good with vectors, think of a vector to the center of the circle c = (4,1) and then a point on the circle itself at r=(x,y). By definition the distance from point r to the center always has the same magnitude. In other words, |r-c| = R, or more conveniently, using the dot product
(r-c)*(r-c) = R^2 = (x-4)^2 + (y -1)^2

Answer 3

Equation of a circle with radius a and centre at the origin (0,0) is x^2 + y^2 = a^2 and since the centre is at another point, we need to shift it.

(X - 4)^2 + (y - 1)^2 = 3^2 would be the equation for the circle given by you.

http://www.analyzemath.com/CircleEq/CircleEq.html

gives a good tutorial about the equation of a circle and allows you to try out various coordinates for the centre and change the radius.