# Equation of Normal to Parabola

Since the equation of the tangent to the parabola

*y*^{2}= 4*ax*at (*x*_{1},*y*_{1}) is*yy*

_{1}= 2

*a*(

*x*+

*x*

_{1}) ...(7)

The slope of the tangent at (

*x*_{1},*y*_{1}) is 2*a*/*y*_{1}. Now the normal at (*x*_{1},*y*_{1}) is perpendicular to the tangent at (*x*_{1},*y*_{1}). Therefore, the slope of normal at (*x*_{1},*y*_{1}) is â€“*y*_{1}/2*a*. Hence the equation of normal at (*x*_{1},*y*_{1}) is*y*â€“

*y*

_{1}= (

*x*â€“

*x*

_{1}) ...(8)

# Normal in parametric form

Replacing

*x*_{1}by*at*^{2}and*y*_{1}by 2*at*in (8), we have*y*â€“ 2*at*= â€“*t*(*x*â€“*at*^{2}) or*y*= â€“*tx*+ 2*at*+*at*^{3}.# Normal in slope form

The equation of normal to the parabola

*y*^{2}= 4*ax**at*(*at*^{2}, 2*at*) is*y*= â€“*tx*+ 2*at*+*at*^{3}. Since*m*is the slope of the normal then*m*= â€“*t*. Then equation of normal is*y*=

*mx*â€“ 2

*am*â€“

*am*

^{3 }...(9)

Thus

*y*=*mx*â€“2*am*â€“*am*^{3}is a normal to the parabola*y*^{2}= 4*ax*where*m*is the slope of the normal.The coordinates of the foot of normal are (

*am*^{2}, â€“2*am*). Comparing (9) with*y*=*mx*+*c*we have*c*= â€“2*am*â€“*am*^{3}, which is condition if*y*=*mx*+*c*is the normal of*y*^{2}= 4*ax*.# Properties of normal

- Normal to the parabola other than axis never passes through focus or focal chord can never be a normal chord.
- Point of intersection of normals at points
*P*â‰¡ (*at*_{1}^{2}, 2*at*_{1}) and*Q*â‰¡ (*at*_{2}^{2}, 2*at*_{2}) to the parabola is*R*â‰¡ [2*a*+*a*(*t*_{1}^{2}+*t*_{2}^{2}+*t*_{1}*t*_{2}), â€“*at*_{1}*t*_{2}(*t*_{1}+*t*_{2})] - Normal at point
*P*(*at*_{1}^{2}, 2*at*_{1}),*y*= â€“*t*_{1}*x*+ 2*at*_{1}+*at*_{1}^{3}meet the parabola again at point Q (*at*_{2}^{2}, 2*at*_{2}) such that*t*_{2}= - Normal at point
*P*(*t*) meets the parabola again at point*Q*at angle tan^{â€“1}.

# Reflection property of parabola

Any ray sent parallel to the axis of parabola passes through the focus of the parabola after reflected by the parabola.