Normal and chord of contact 

            In this page 'Normal and chord of contact',  we are going to see how to derive the equation of normal of ellipse and equation of chord of contact of ellipse.

Equation of normal to the given ellipse:

                The equation of the tangent at (x₁,y₁) is

                                   xx₁/a²  +  yy₁/b²     = 1

                                            y  =  -b²/a² . xx₁/y₁ + b²/y₁

                The slope of the tangent is -b²/a² . x₁/y₁

                Since the normal is perpendicular to the tangent,

                 The slope of the normal is a²/b² . y₁/x₁

                  The equation of normal at the point (x₁,y₁) is

                                         y-y₁    =   a²/b² . y₁/x₁ . (x-x₁) 

                  Simplifying the equation,

                                      a²x/x₁  -  b²y/y₁  =  a²-b²

                  This is the required equation of normal of the given ellipse.


           Two tangents can be drawn from any point (x₁,y₁) to the ellipse

x²/a² + y²/b² = 1.

            We know that the line y=mx+c be a tangent to the ellipse is

 c= ±√( a²m² +  b²).

             Let (x₁,y₁)  lie on the line.

                                    y₁  =    mx₁ +√( a²m² +  b²)

                              y₁-mx₁ =   √( a²m² +  b²)

             Squaring on both sides

                           (y₁-mx₁)² =   a²m² + b²

                           m²(x₁²-a²)-2x₁y₁m+y₁²-b² = 0

             which is a quadratic equation in m, so we will get two roots which implies that two tangents can be drawn from the point (x₁,y₁).

Equation of chord of contact:

               Let the tangents from the point P(x₁,y₁) touch the ellipse at

Q(x₂,y₂) and R(x₃,y₃).

               The equation of tangent at Q(x₂,y₂) is xx₂/a² + yy₂/b² = 1.

               P lies on this line also. So,

                                     x₁x₂/a² + y₁y₂/b² = 1 ------------- I

               The equation of tangent at R(x₃,y₃) is

                                     xx₃/a²  +  yy₃/b²  =  1

               P lies on this line also. So,

                                     x₁x₃/a² +  y₁y₃/b² =   1------------II

               From equation I and II it is obvious that the points of contact at Q(x₂,y₂) and R(x₃,y₃) lies on the line xx₁/a² + yy₁/b² = 1.

               So  xx₁/a² + yy₁/b² = 1 is the equation of chord of contact.



An important property of ellipse:

                        If P is any point on the ellipse whose foci are S and S'. Then SP + S'P =2a where 2a is the major axis.


                    Let X axis is the major axis and origin is the centre of the ellipse x²/a² + y²/b² = 1.

                   Let 'e' be the eccentricity of the ellipse.

                   From the diagram SP =ePM

                                             S'P = ePM'

                                       SP+S'P = e(PM+PM')

                                                   = e(MM')

                                                   = 2eCZ

                                                   = 2e a/e

                                                   = 2a.





          Parents and teachers can guide the students to go through the derivations given in this page 'Normal and chord of contact' step by step. Students can follow the steps and try to derive the equations on their own. If you have any doubt you can contact us through mail, we will help you to clear your doubts. 



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