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BodhiAI
Nov. 15, 2019

Equation of Tangent in various forms:

 

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

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

Hence, the equation 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 obtained by substituting the parametric coordinates in the equation of tangent in Point Form.

Coordinates of Point of Contact

Let 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