Equation of Normal

(i) Point form :

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

Differentiate w.r.t.x

$\begin{array}{rl}& 2y\frac{dy}{dx}=4a\\ & \frac{dy}{dx}=\frac{2a}{y}\end{array}$

Slope of normal $=-\frac{{\mathrm{y}}_1}{2\mathrm{a}}$

Equation of normas at $({\mathrm{x}}_1,{\mathrm{y}}_1)$ is

$y-y_1=-\frac{y_1}{2a}(x-x_1)$

(ii) Parametric form :

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

replace $x_1$ by ${\mathrm{at}}^2$ and ${\mathrm{y}}_1$ by 2 at

$\begin{array}{rl}& y-2at=-\frac{2at}{2a}(x-a{t}^2)\\ & y=-tx+a{t}^3+2at\\ & tx+y-a{t}^3-2at=0\end{array}$

(iii) Slope form:

$\begin{array}{rl}& \text{ Replace }t\text{ by }-m\\ & y=mx-2am-a^3\end{array}$

$y=mx+c$ is normal to parabola $y^2=4ax$ 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

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

Properties of Normal

1. If the normal at the point $\mathrm{P}({\mathrm{at}}_1{}_1^2,2{\mathrm{at}}_1)$ meets the parabola at

$\mathrm{Q}({\mathrm{at}}_2{}^2,2{\mathrm{at}}_2)$, 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

$\begin{array}{rl}& 2{\mathrm{at}}_2=-{\mathrm{at}}_1{\mathrm{t}}_2{}^2+2{\mathrm{at}}_1+{\mathrm{at}}_1{}^3\\ & 2\mathrm{a}({\mathrm{t}}_2-{\mathrm{t}}_1)=-{\mathrm{at}}_1({\mathrm{t}}_2{}^2-{\mathrm{t}}_1^2)\\ & 2=-{\mathrm{t}}_1({\mathrm{t}}_2+{\mathrm{t}}_1)\\ & \therefore {\mathrm{t}}_2=-{\mathrm{t}}_1-\frac{2}{{\mathrm{t}}_1}\end{array}$

2. If the normal at the points $(a{t}_1{}^2,2{\mathrm{at}}_1)$ and $({\mathrm{at}}_2{}^2,2{\mathrm{at}}_2)$ meet on the parabola ${\mathrm{y}}^2=4\mathrm{ax}$, then ${\mathrm{t}}_1{\mathrm{t}}_2=2$.

Let the equation of normal at $({\mathrm{at}}_1{}^2,2{\mathrm{at}}_1)$ and $({\mathrm{at}}_2{}_2^2,2{\mathrm{at}}_2)$ 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 $({\mathrm{at}}_3{}^2,2{\mathrm{at}}_3)$ then

$\begin{array}{rl}& {\mathrm{t}}_3=-{\mathrm{t}}_1-\frac{2}{{\mathrm{t}}_1}\text{ and }{\mathrm{t}}_3=-{\mathrm{t}}_2-\frac{2}{{\mathrm{t}}_2}\\ & \therefore -{\mathrm{t}}_1-\frac{2}{{\mathrm{t}}_1}=-{\mathrm{t}}_2-\frac{2}{{\mathrm{t}}_2}\\ & {\mathrm{t}}_2-{\mathrm{t}}_1=2\left(\frac{1}{t_1}\frac{1}{t_2}\right)\Rightarrow t_2-t_1=2\left(\frac{t_2-t_1}{t_1{t}_2}\right)\\ & \Rightarrow {\mathrm{t}}_1{\mathrm{t}}_2=2\end{array}$

3. No normal other than axis passes through focus.

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

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

$\begin{array}{rl}\therefore & 0=ma-2am-a^3\\ & 0=-am-a^3\\ & 0=-am((1+m^2)\Rightarrow m=0\text{ i.e. axis }\\ & 1+{\mathrm{m}}^2=0\text{ which is not possible. }\end{array}$

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) $x+4y=63$

(c) $3x-y=33$

(d) $y+3x=51$

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

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

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

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

• 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 $\mathrm{S}$. $(\mathrm{SP}{)}^2=\mathrm{SA}\times \mathrm{SB}$

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

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

• 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 $y^2=8x$ 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 $y^2=4ax$, 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 $y^2=4x$ 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 $y^2=4x$ and contained in it, is

(a). 2

(b). 4

(c). 6

(d). 8

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

(a). $3$

(b). $1$

(c). $2$

(d). $-3$

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

(a). $2am+a^3$

(b).

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

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