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Oct. 31, 2019

Circle:

Condition of Tangency

Le let the line y = mx + c is tangent to the circle x2 + y2 = a2 then

which is a quadratic in x, then according to Condition of Tangency, we must have

which is the condition for which the line y = mx+c is tangent to the circle

Equation of Tangent in Different Forms

A. Point Form

Let is any point on the circle then the equation of tangent through this point P (as point of contact) is given by the equation

In general, if the equation of the circle is and is any point on S, then the equation of tangent to S through P is given by the ation

B. Slope Form

Let m be the slope of tangent to the circle

Since, m is the slope of tangent, let us assume a general equation of tangent line having slope m.

Now, according to Condition of Tangency, the line y = mx+c is tangents to circle if

                                        of tangent(s) to the circle having slope m is/are

C. Parametric Form

Let are the coordinates of any point on the circle having radius a in parametric form. Then the equation of tangent through P can be btained by substituting the parametric coordinates in the equation of tangent in Point Form.

                                          c                                                                                                                                                               ordinates of Point of Contact      

is the equation of circle, then the equation of tangents to the circle having slope m is given by the eqation 

 

......(1)

Solving the equation of line and the circle, we will obtain the coordinates of the point satisfying both the line and the circle i.e. the coordinates of point of contact, such that

                  

               

                

                      

and {Substituting respective value of x in equation (1)}

Hence, the coordinates of point of contact are