Chord of Contact

Let $\mathrm{PA}$ and $\mathrm{PB}$ be tangents drawn through the point $\mathrm{P}(\mathrm{h},\mathrm{k})$.

Equation of tangent at $\mathrm{A}$ is

${\mathrm{yy}}_1=2\mathrm{a}(\mathrm{x}+{\mathrm{x}}_1)$

Equation of tangent at $B$ is

${\mathrm{yy}}_2=2\mathrm{a}(\mathrm{x}+{\mathrm{x}}_2)$

Both lines pass through $(\mathrm{h},\mathrm{k})$

$\begin{array}{rl}& \therefore {\mathrm{ky}}_1=2\mathrm{a}(\mathrm{h}+{\mathrm{x}}_1)\dots \dots .(1)\\ & {\mathrm{ky}}_2=2\mathrm{a}(\mathrm{h}+{\mathrm{x}}_2)\dots \dots \dots .(2)\end{array}$

Hence $\mathrm{A}({\mathrm{x}}_1,{\mathrm{y}}_1)$ and $\mathrm{B}({\mathrm{x}}_2,{\mathrm{y}}_2)$ lie on $\mathrm{ky}=2\mathrm{a}(\mathrm{x}+\mathrm{h})$ or $\mathrm{T}=0$ ie. equation of chord $\mathrm{AB}$.

Equation of chord whose midpoint $({\mathrm{x}}_1,{\mathrm{y}}_1)$ is given :

$\begin{array}{rl}& {\mathrm{S}}_{\equiv {\mathrm{y}}^2-4\mathrm{ax}}\\ & {\mathrm{S}}_1\equiv {\mathrm{y}}_1^2-4\mathrm{ax}\\ & \mathrm{T}\equiv {\mathrm{y}}_1-2\mathrm{a}(\mathrm{x}+{\mathrm{x}}_1)\end{array}$

Equation of $\mathrm{AB}$ is $\mathrm{T}=\mathrm{S}$

Reflection Property of Parabola

Module - 7

Let $PQ$ be tangent of the parabola $y^2=ax$ at point $P(a{t}^2,2at)$ Equation of tangent at $\mathrm{P}$ is $\mathrm{ty}=\mathrm{x}+{\mathrm{at}}^2$

$\mathrm{Q}$ is the point of intersection of tangent and $\mathrm{x}$-axis

So $\mathrm{Q}(-{\mathrm{at}}^2,0)$

$\therefore \mathrm{SQ}=\mathrm{SA}+\mathrm{AQ}=\mathrm{a}+{\mathrm{at}}^2=\mathrm{a}(1+{\mathrm{t}}^2)$

$\mathrm{SP}=\mathrm{PM}={\mathrm{at}}^2+\mathrm{a}=\mathrm{a}(1+{\mathrm{t}}^2)$

$\therefore \mathrm{SP}=\mathrm{SQ}$

$\mathrm{\angle }\mathrm{SPQ}=\mathrm{\angle }\mathrm{SQP}$

Also, $\mathrm{\angle }\mathrm{MPQ}=\mathrm{\angle }\mathrm{PQS}$ (alternate angles)

Hence $\mathrm{\angle }\mathrm{MPQ}=\mathrm{\angle }\mathrm{QPS}$

Also $\mathrm{\angle }\mathrm{IPN}=\mathrm{\angle }\mathrm{NPS}$

So PN normal bisects $\mathrm{\angle }$ IPS

Therefore PI is incident ray then PS is reflected ray. So any ray incident parallel to axis of the parabola after reflection it passes through focus.

Examples

1. If three normals can be drawn to the parabola $y^2-2y=4x-9$ from the point $(a,b)$, then the range of the value of a is

(a). $(2,\mathrm{\infty })$

(b). $(1,\mathrm{\infty })$

(c). $(-\mathrm{\infty },-2)$

(d). $(4,\mathrm{\infty })$

Show Answer

Solution: ${\mathrm{y}}^2-2\mathrm{y}=4\mathrm{x}-9$

$(y-1{)}^2=4(x-2)$

For real three normal’s

as $\mathrm{h}>2\mathrm{a}$

$a>4$

Answer: d

2. A circle and a parabola ${\mathrm{y}}^2=4\mathrm{ax}$ intersect at four points. The algebraic sum of the ordinates of the four points is

(a). $0$

(b). $1$

(c). $-1$

(d). $4$

3. Maximum number of common normals of ${\mathrm{y}}^2=4\mathrm{ax}$ and ${\mathrm{x}}^2=4$ by is

(a). 3

(b). 4

(c). 6

(d). 5

4. The curve described parametrically by $\mathrm{x}={\mathrm{t}}^2+\mathrm{t}+1,\mathrm{y}={\mathrm{t}}^2-\mathrm{t}+1$ represents

(a). a pair of straight lines

(b). an ellipse

(c). a parabola

(d). a hyperbola

5. If $\mathrm{x}+\mathrm{y}=\mathrm{k}$ is normal to ${\mathrm{y}}^2=12\mathrm{x}$, then $\mathrm{k}$ is

(a). $3$

(b). $9$

(c). $-9$

(d). $-3$

6. The equation of the common tangent touching the circle $(x-3{)}^2+y^2=9$ and the parabola ${\mathrm{y}}^2=4\mathrm{x}$ above the $\mathrm{x}$-axis is

(a). $\sqrt{3}y=3x+1$

(b). $\sqrt{3}y=-(x+3)$

(c). $\sqrt{3}y=x+3$

(d). $\sqrt{3}y=-(3x+1)$