Equation of Normal

(i) Point form :

$\mathrm{y}^{2}=4 \mathrm{ax}$

Differentiate w.r.t.x

$ \begin{aligned} & 2 y \frac{d y}{d x}=4 a \\ & \frac{d y}{d x}=\frac{2 a}{y} \end{aligned} $

Slope of normal $=-\frac{\mathrm{y} _{1}}{2 \mathrm{a}}$

Equation of normas at $\left(\mathrm{x} _{1}, \mathrm{y} _{1}\right)$ is

$ y-y _{1}=-\frac{y _{1}}{2 a}\left(x-x _{1}\right) $

(ii) Parametric form :

$\mathrm{P}\left(\mathrm{at}^{2}, 2 \mathrm{at}\right)$

replace $x _{1}$ by $\mathrm{at}^{2}$ and $\mathrm{y} _{1}$ by 2 at

$ \begin{aligned} & y-2 a t=-\frac{2 a t}{2 a}\left(x-a t^{2}\right) \\ & y=-t x+a t^{3}+2 a t \\ & t x+y-a t^{3}-2 a t=0 \end{aligned} $

(iii) Slope form:

$ \begin{aligned} & \text { Replace } t \text { by }-m \\ & y=m x-2 a m-a^{3} \end{aligned} $

$y=m x+c$ is normal to parabola $y^{2}=4 a x$ if

$\mathrm{c}=-2 \mathrm{am}-\mathrm{am}^{3}$ ie., condition of normal.

Equation of Normal

Equation of normal to the parabola $(\mathrm{y}-\mathrm{k})^{2}=4(\mathrm{x}-\mathrm{h})$ is

$ y-k=m(x-h)-2 a m-a m^{3} $

Properties of Normal

1. If the normal at the point $\mathrm{P}\left(\mathrm{at} _{1}{ } _{1}^{2}, 2 \mathrm{at} _{1}\right)$ meets the parabola at

$\mathrm{Q}\left(\mathrm{at} _{2}{ }^{2}, 2 \mathrm{at} _{2}\right)$, then $\mathrm{t} _{2}=-\mathrm{t} _{1}-\frac{2}{\mathrm{t} _{1}}$

Let equation of parabola be $\mathrm{y}^{2}=4 \mathrm{ax}$.

Equation of normal at $\mathrm{P}$ is

$ \mathrm{y}=-\mathrm{t} _{1} \mathrm{x}+2 \mathrm{at} _{1}+\mathrm{at} _{1}^{3} $

Point $\mathrm{Q}$ lies on the normal, so

$ \begin{aligned} & 2 \mathrm{at} _{2}=-\mathrm{at} _{1} \mathrm{t} _{2}{ }^{2}+2 \mathrm{at} _{1}+\mathrm{at} _{1}{ }^{3} \\ & 2 \mathrm{a}\left(\mathrm{t} _{2}-\mathrm{t} _{1}\right)=-\mathrm{at} _{1}\left(\mathrm{t} _{2}{ }^{2}-\mathrm{t} _{1}^{2}\right) \\ & 2=-\mathrm{t} _{1}\left(\mathrm{t} _{2}+\mathrm{t} _{1}\right) \\ & \therefore \mathrm{t} _{2}=-\mathrm{t} _{1}-\frac{2}{\mathrm{t} _{1}} \end{aligned} $

2. If the normal at the points $\left(a t _{1}{ }^{2}, 2 \mathrm{at} _{1}\right)$ and $\left(\mathrm{at} _{2}{ }^{2}, 2 \mathrm{at} _{2}\right)$ meet on the parabola $\mathrm{y}^{2}=4 \mathrm{ax}$, then $\mathrm{t} _{1} \mathrm{t} _{2}=2$.

Let the equation of normal at $\left(\mathrm{at} _{1}{ }^{2}, 2 \mathrm{at} _{1}\right)$ and $\left(\mathrm{at} _{2}{ } _{2}^{2}, 2 \mathrm{at} _{2}\right)$ be

$ \mathrm{y}=-\mathrm{t} _{1} \mathrm{x}+2 \mathrm{at} _{1}+\mathrm{at} _{1}^{3} $

and $y=-\mathrm{t} _{2} \mathrm{x}+2 \mathrm{at} _{2}+\mathrm{at} _{2}{ }^{3}$

meet the parabola $\mathrm{y}^{2}=4 \mathrm{ax}$ at $\left(\mathrm{at} _{3}{ }^{2}, 2 \mathrm{at} _{3}\right)$ then

$ \begin{aligned} & \mathrm{t} _{3}=-\mathrm{t} _{1}-\frac{2}{\mathrm{t} _{1}} \text { and } \mathrm{t} _{3}=-\mathrm{t} _{2}-\frac{2}{\mathrm{t} _{2}} \\ & \therefore-\mathrm{t} _{1}-\frac{2}{\mathrm{t} _{1}}=-\mathrm{t} _{2}-\frac{2}{\mathrm{t} _{2}} \\ & \mathrm{t} _{2}-\mathrm{t} _{1}=2\left(\frac{1}{t _{1}} \frac{1}{t _{2}}\right) \Rightarrow t _{2}-t _{1}=2\left(\frac{t _{2}-t _{1}}{t _{1} t _{2}}\right) \\ & \Rightarrow \mathrm{t} _{1} \mathrm{t} _{2}=2 \end{aligned} $

3. No normal other than axis passes through focus.

Let equation of normal be $y=m x-2 a m-a m^{3}$

passes through $(a, 0)$ ie. focus

$ \begin{aligned} \therefore \quad & 0=m a-2 a m-a^{3} \\ & 0=-a m-a^{3} \\ & 0=-a m\left(\left(1+m^{2}\right) \Rightarrow m=0\right. \text { i.e. axis } \\ & 1+\mathrm{m}^{2}=0 \text { which is not possible. } \end{aligned} $

Example: 1 Three normals to $\mathrm{y}^{2}=4 \mathrm{x}$ pass through the point $(15,12)$. One of the normals is

(a) $x+y=27$

(b) $x+4 y=63$

(c) $3 x-y=33$

(d) $y+3 x=51$

Example: 2 The minimum distance between the curves $x^{2}+y^{2}-12 x+31=0$ and $y^{2}=4 x$ is

(a) $\sqrt{5}$

(b) $2 \sqrt{5}$

(c) $6-\sqrt{5}$

(d) None of these.

Important Properties :

• If the tangent and normal at any point ’ $\mathrm{P}$ ’ of the parabola intersect the axis at $\mathrm{T}$ and $\mathrm{N}$ then $\mathrm{ST}$ $=\mathrm{SN}=\mathrm{SP}$ where $\mathrm{S}$ in the focus.

• The portion of a tangent to a parabola cut off between the directrix & the curve subtends a right angle at the focus.

$ \angle \mathrm{PSQ}=90^{\circ} $

• Any tangent to a parabola and the perpendicular on it from the focus meet on the tangent at the vertex.

$\angle \mathrm{PQS}=90^{\circ}$

• If the tangents at $\mathrm{A}$ and $\mathrm{B}$ meet in $\mathrm{P}$ then $\mathrm{PA}$ and $\mathrm{PB}$ subtends equal angles at the focus $\mathrm{S}$. $(\mathrm{SP})^{2}=\mathrm{SA} \times \mathrm{SB}$

$\triangle \mathrm{SAP} \sim \triangle \mathrm{SPB}$

$\angle \mathrm{PSA}=\angle \mathrm{PSB}$.

• The area of the triangle formed by three points on a parabola is twice the area of the triangle formed by the tangents at these points.

Practice questions

1. The normal at the point $(2,4)$ of the parabola $y^{2}=8 x$ meets the parabola at the point

(a). $(18,-12)$

(b). $(12,-18)$

(c). $(12,18)$

(d). $(18,12)$

2. If $y+b=m _{1}(x+a)$ and $y+b=m _{2}(x+a)$ are two tangents to the parabola $y^{2}=4 a x$, then

(a). $\mathrm{m} _{1} \mathrm{~m} _{2}=1$

(b). $\mathrm{m} _{1} \mathrm{~m} _{2}=-1$

(c). $\mathrm{m} _{1}+\mathrm{m} _{2}=0$

(d). $\mathrm{m} _{1}-\mathrm{m} _{2}=0$

3. If normals at the ends of the double ordinate $x=4$ of parabola $y^{2}=4 x$ meet the curve again in $\mathrm{P}$ and $\mathrm{P}^{\prime}$ respectively, then $\mathrm{P}=$

(a). 10

(b). 6

(c). 12

(d). 18

4. Radius of the largest circle which passes through the focus of the parabola $y^{2}=4 x$ and contained in it, is

(a). 2

(b). 4

(c). 6

(d). 8

5. If the normal at $(1,2)$ on the parabola $y^{2}=4 x$ meets the parabola again at the point $\left(t^{2}, 2 t\right)$, then $t$ is

(a). $3$

(b). $1$

(c). $2$

(d). $-3$

6. If $x=m y+c$ is a normal to the parabola $\frac{x^{2}-\mathrm{a}}{4}+\frac{\mathrm{a}}{\mathrm{a} m \text { Then }} \mathrm{c}=$

(a). $2 a m+a^{3}$

(b).

(c). $-2 \mathrm{am}-\mathrm{am}^{3}$

(d). $-\frac{2 \mathrm{a}}{\mathrm{m}}-\frac{\mathrm{a}}{\mathrm{m}^{3}}$