# Normal to a Circle at a Given Point

The normal of a circle at any point is a straight line which is perpendicular to the tangent at the point of contact.

The normal of the circle always passes through the center of the circle.

To find the equation of normal to the circle,

*x*^{2}+*y*^{2}+ 2*gx*+ 2*fy*+*c*= 0 at the point (*x*_{1},*y*_{1}) on it.Since normal passes through the center we have slope of normal

*CP*= . Hence equation of normal is*y*â€“

*y*

_{1}= (

*x*â€“

*x*

_{1})

or

# Chord of contact

From any external point

*A*(*x*_{1},*y*_{1}) draw pair of tangents*AP*and*AQ*touching the circle at*P*(*x*â€²,*y*â€²) and*Q*(*x*â€³,*y*â€³) respectively. The line joining*P*and*Q*is called chord of contact and its equation is given by*xx*_{1}+*yy*_{1}=*a*^{2}or*T*= 0, where*T*=*xx*_{1}+*yy*_{1}â€“*a*^{2}.Equation of the chord bisected at a given point Let any chord

*AB*of the circle*x*^{2}+*y*^{2}+ 2*gx*+ 2*fy*+*c*= 0 be bisected at*D*(*x*_{1},*y*_{1}).Then its equation is

*xx*_{1}+*yy*_{1}+*g*(*x*+*x*_{1}) +*f*(*y*+*y*_{1}) +*c*=*x*_{1}^{2}+*y*_{1}^{2}+ 2*gx*_{1}+ 2*fy*_{1}+*c*or*T*=*S*_{1}.**The chord bisected at point (**

*Note:**x*

_{1},

*y*

_{1}) is the chord of minimum length passing through the point (

*x*

_{1},

*y*

_{1}) and at maximum distance from the center.

# Pair of tangents

Let the circle be

*x*^{2}+*y*^{2}=*a*^{2}. Let the given external point be*P*(*x*_{1},*y*_{1}).From point

*P*(*x*_{1},*y*_{1}) two tangents*PT*and*PR*can be drawn to the circle, touching the circle at*T*and*R*respectively.Then equation of pair of tangents is (

*x*^{2}+*y*^{2}â€“*a*^{2}) (*x*_{1}^{2}+*y*_{1}^{2}â€“*a*^{2}) = (*xx*_{1}+*yy*_{1}â€“*a*^{2})^{2}or*SS*_{1}=*T*^{2}, where*S*=*x*^{2}+*y*^{2}â€“*a*^{2},*S*_{1}=*x*_{1}^{2}+*y*_{1}^{2}â€“*a*^{2}, and*T*=*xx*_{1}+*yy*_{1}â€“*a*^{2}.# Director circle

The locus of the point of intersection of two perpendicular tangents to a given circle is a circle known as its director circle and is given by

*x*^{2}+*y*^{2}= 2*a*^{2}.Equation of director circle for circle (

*x*â€“*p*)^{2}+ (*y*â€“*q*)^{2}=*a*^{2}is given by (*x*â€“*p*)^{2}+ (*y*â€“*q*)^{2}= 2*a*^{2}.