Curves - Circles

 

Definition:   Given a fixed point, a circle is a collection of points with the same distance to that fixed point on the same plane. That fixed point is known as the center of the circle and the distance is known as the radius of the circle.

 

                  We formalize the expression by using the mathematical language as follows:

                             Fixed point :  O(x0, y0)       radius  r

                             From the distance formula, we know that

 

                                             ((x-x0)² + (y-y0)²)^1/2 = r

 

                           It is equivalent to

 

                                                (x-x0)² + (y-y0)² = r²

 

 

Theorem (  Tangent Line Equation of Circle ): 

                    Given a point A(a,b) on the circle C:  (x-x0)² + (y-y0)² = r² ,   the tangent line passing A is

 

                                       L:     (a-x0)x + (b-y0)y = (a-x0)a + (b-y0)b

 

 

       Proof:

                        From the equation of the circle,  the center O is   (x0, y0) .

 

                        And the tangent line crosses the circle at point A  

                        Therefore,    is perpendicular to the line.

                        Then   is a normal vector to the tangent line.

 

                        So, the tangent line equation is of the form

 

(a-   x0)x + (b-y0)y = c

 

                         A(a,b) is on this line, it must satisfy the line equation.  We use it to find c :

 

(a-   x0)a + (b-y0)b =c

 

                          Hence,      the tangent line equation is

 

(a-   x0)x + (b-y0)y = (a-x0)a + (b-y0)b

 

 

 

 

              Short Cut:

 

                        From the result of Calculus,  we can use the derivative of implicit function:

 

                                               (x-x0)² + (y-y0)² = r²

                                                       2(x-x0) + 2(y-y0)  = 0

                              is the function for the slope of the tangent line at (x,y) on the curve .

 

                              In this case,  the slope of the tangent line at (a,b) satisfies

                                              

                                                       2(a-x0) + 2(b-y0) m = 0

 

                                               where m is the slope of the tangent line.