PARABOLA-8
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}\left(\mathrm{x}+\mathrm{x} _{1}\right) $
Equation of tangent at $B$ is
$ \mathrm{yy} _{2}=2 \mathrm{a}\left(\mathrm{x}+\mathrm{x} _{2}\right) $
Both lines pass through $(\mathrm{h}, \mathrm{k})$
$\begin{aligned} & \therefore \mathrm{ky}_1=2 \mathrm{a}\left(\mathrm{h}+\mathrm{x}_1\right) …….(1)\\ & \mathrm{ky}_2=2 \mathrm{a}\left(\mathrm{h}+\mathrm{x}_2\right)……….(2)\end{aligned}$
Hence $\mathrm{A}\left(\mathrm{x} _{1}, \mathrm{y} _{1}\right)$ and $\mathrm{B}\left(\mathrm{x} _{2}, \mathrm{y} _{2}\right)$ 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 $\left(\mathrm{x} _{1}, \mathrm{y} _{1}\right)$ is given :
$ \begin{aligned} & \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}\left(\mathrm{x}+\mathrm{x} _{1}\right) \end{aligned} $
Equation of $\mathrm{AB}$ is $\mathrm{T}=\mathrm{S}$
Reflection Property of Parabola
Module - 7
Let $P Q$ be tangent of the parabola $y^{2}=a x$ at point $P\left(a t^{2}, 2 a t\right)$ 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 $\mathrm{Q}\left(-\mathrm{at}^{2}, 0\right)$
$\therefore \mathrm{SQ}=\mathrm{SA}+\mathrm{AQ}=\mathrm{a}+\mathrm{at}^{2}=\mathrm{a}\left(1+\mathrm{t}^{2}\right)$
$\mathrm{SP}=\mathrm{PM}=\mathrm{at}^{2}+\mathrm{a}=\mathrm{a}\left(1+\mathrm{t}^{2}\right)$
$\therefore \mathrm{SP}=\mathrm{SQ}$
$\angle \mathrm{SPQ}=\angle \mathrm{SQP}$
Also, $\angle \mathrm{MPQ}=\angle \mathrm{PQS}$ (alternate angles)
Hence $\angle \mathrm{MPQ}=\angle \mathrm{QPS}$
Also $\angle \mathrm{IPN}=\angle \mathrm{NPS}$
So PN normal bisects $\angle$ 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 $y^{2}-2 y=4 x-9$ from the point $(a, b)$, then the range of the value of a is
(a). $(2, \infty)$
(b). $(1, \infty)$
(c). $(-\infty,-2)$
(d). $(4, \infty)$
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). $4$
Show Answer
Solution: Let equation of circle be $\mathrm{x}^{2}+\mathrm{y}^{2}+2 \mathrm{gx}+2 \mathrm{fy}+\mathrm{C}=0$
and equation of parabola is $x=\mathrm{at}^{2}, \mathrm{y}=2 \mathrm{at}$
Solving it, we get
$ \begin{aligned} & a^{2} t^{4}+4 a^{2} t^{2}+2 a g t^{2}+4 a f t+c=0 \\ & a^{2} t^{4}+2 a(2 a+g) t^{2}+4 a f t+c=0 \\ & t _{1}+t _{2}+t _{3}+t _{4}=0 \\ & 2 a _{1}+2 a _{2}+2 a _{3}+2 a _{4}=0 \\ & y _{1}+y _{2}+y _{3}+y _{4}=0 \end{aligned} $
$ \text { Or }2 \mathrm{at} _{1}+2 \mathrm{at} _{2}+2 \mathrm{at} _{3}+2 \mathrm{at} _{4}=0 $
Answer: a
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
Show Answer
Solution: Normals of $\mathrm{y}^{2}=4 \mathrm{ax}$ and $\mathrm{x}^{2}=4$ by in slope form are
$\mathrm{y}=\mathrm{mx}-2 \mathrm{am}-\mathrm{am}^{2}$
$y=m x+2 b+\frac{b}{m^{2}}$
For common normal $2 b+\frac{b}{m^{2}}=-2 a m-a^{2}$
$ \mathrm{am}^{4}+2 \mathrm{am}^{3}+2 \mathrm{bm}^{2}+\mathrm{b}=0 $
This mean there can be atmost 4 common normals
Answer: b
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
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Solution: $x+y=2\left(t^{2}+1\right)$ and $x-y=2 t$
$\therefore \frac{\mathrm{x}+\mathrm{y}}{2}=\frac{\mathrm{x}-\mathrm{y}}{2}^{2}+1$
$\mathrm{x}^{2}+\mathrm{y}^{2}-2 \mathrm{xy}+4=2 \mathrm{x}+2 \mathrm{y}$
$x^{2}+y^{2}-2 x y-2 x-2 y+4=0$
$\Delta=\left|\begin{array}{rrr}1 & -1 & -1 \\ -1 & 1 & -1 \\ -1 & -1 & 4\end{array}\right|=3-5-2=-4 \neq 0$
$\mathrm{h}^{2}-\mathrm{ab}=1-1=0 \quad \therefore$ It is a parabola
Answer: c
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). $-3$
Show Answer
Solution: Normal to $\mathrm{y}^{2}=12 \mathrm{x}$ is
$ \begin{aligned} & \mathrm{m}=-1 \\ & \mathrm{y}=-\mathrm{x}+6+3 \\ & \mathrm{x}+\mathrm{y}=9 \end{aligned} $
Answer: b
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=3 x+1$
(b). $ \sqrt{3} y=-(x+3)$
(c). $ \sqrt{3} y=x+3$
(d). $\sqrt{3} y=-(3 x+1)$
Show Answer
Solution: Equation of tangent to the parabola is $\mathrm{y}=\mathrm{mx}+\frac{1}{\mathrm{~m}}$ and equation of tangent to the circle is $y=m(x-3) \pm 3 \sqrt{1+\mathrm{m}^{2}}$ both the equation are identical i.e. $\quad \frac{1}{m}=-3 m \pm 3 \sqrt{1+m^{2}}$
$ \begin{gathered} \frac{1}{m}+3 m^{2}=3^{2}\left(1+m^{2}\right) \\ \frac{1}{m^{2}}+9 m^{2}+6=9+9 m^{2} \\ 1=3 m^{2} \quad m= \pm \frac{1}{\sqrt{3}} \end{gathered} $
Equation of common tangent is $\sqrt{3} y=x+3$ (tangent lying above $x$-axis)
Answer: c
