**What is a parabola**

Parabola is a conic section defined as, Locus of a point which moves such that its distance from a fixed point is always in a constant ratio to its perpendicular distance from a fixed line, is called a conic section. $$\frac{PS}{PM}=e\,\,\,\,\,(Constant)$$ where ‘e’ is called as eccentricity of conic section , e = 1 for the parabola.

We can now redefine the parabola as. Locus of a point which moves such that its distance from a fixed point to its perpendicular distance from a fixed line is equal ( i.e e = 1 ).

**Equation of parabola **

$$Since\,\,\,\,SP^2=PM^2$$ $$(h-a)^2+k^2=(h+a)^2$$ $$\Rightarrow k^2=4ah $$

Now replace h and k with x and y respectivelly. $$y^2=4ax$$

The equation is equivalent to geometrical proposition that $$PN^2=4(AS)(AN)$$

Therefore equation of parabola is $$PN=\frac{\vert px+qy+r \vert}{\sqrt{p^2+q^2}}$$ $$ AS = a \,\,\,, AN = PM=\frac{\vert qx-py+\alpha \vert}{\sqrt{p^2+q^2}}$$ $$ \therefore \,\,\frac{(px+qy+r)^2}{p^2+q^2}=4a\frac{\vert qx-py+\alpha \vert}{\sqrt{p^2+q^2}}$$

represent the equation of parabola whose equation of axis is px + qy + r = 0 and tangent at vertex is qx – px + $\alpha$ = 0 .

**Equation of parabola given coordinates of focus and equation of directrix.**

let S($\,\alpha\,,\,\beta\,$) be the focus and ax+by+c=0 be the equation of directrix.

Equation of parabola is $$PS=PM$$ $$\sqrt{(x-\alpha)^2+(y-\beta)^2}=\frac{\vert ax+by+c \vert}{\sqrt{a^2+b^2}}$$ $$ (x-\alpha)^2+(y-\beta)^2=\frac{(ax+by+c)^2}{a^2+b^2} $$

**Parametric coordinates on different standard parabola**

**Equation of Parabola**

**X coordinate**

**Y coordinate**

$y^2=4ax$

$x=at^2$

y=2at

$x^2=4ay$

x=2at

$y=at^2$

$y^2=-4ax$

$x=-at^2$

y=2at

$x^2=-4ay$

x=-2at

$y=-at^2$

**Practice questions on formation of equation of parabola**

**Question 1 :-$$ $$Find locus of centre of a variable circle , which intercepts a chord of fixed length 2a on the axis of x and passes through a fixed point on the y axis (0,b).**

**Question 2 :-$$ $$Find locus of a variable point P such that length of tangent from point P to a circle $x^2+y^2$=1 is equal to distance of point P from the line x+2=0.**

**Question 3 :- $$ $$ Two parabolas have a common axis and concavities in opposite directions . If a line parallel to the axis of parabolas at A and B . Then prove that the locus of the middle point of AB is a parabola . If given parabolas are not congurant parabolas.**

**Focal Chord of parabola**

**A chord to parabola passing through the focus is called as focal chord of parabola**

(1) PQ is a variable focal chord where point P $(at_1^2,2at_1)$ and Q $(at_2^2,2at_2)$ then $$ Slope\,of\,PQ=Slope\,of\,PS $$ $$\frac{2}{t_1+t_2}=\frac{2t_1}{t_1^2-1}$$ $$t_1^2-1=t_1^2+t_1t_2$$ $$t_1t_2=-1$$

(2) Length of focal chord in terms of parameter $t_1$ and $t_2$.

Focal distance of point p($t_1$) , PS = a$(1+t_1^2)$

Similarly Focal distance of point Q($t_2$) , PQ = a$(1+t_2^2)$ = a$(1+\frac{1}{t_1^2})$

Therefore Focal length PQ = PS + PS = a ($t_1^2+\frac{1}{t_1^2}$+2) $$PQ=a(t_1+\frac{1}{t_1})^2\ge 4a $$

Length of focal chord PQ in term of $\theta$

Let PS = $r_1$ , QS = $r_2$

Any point on PQ with respect to focus S( a , 0 ) is ( x , y ) = ( r Cos$\theta$ + a , rSin$\theta$ )

for r = $r_1$ and $r_2$ lie on the parabola.

Hence $$ (rSin\theta)^2=4a(rCos\theta + a)$$ $$(Sin^2\theta) r^2-(4aCos\theta) r -4a^2 = 0 $$ $$\Rightarrow r_1+r_2 = \frac{4aCos\theta}{Sin^2\theta} \,\,\,\,\,;\,\,\,\,\, r_1.r_2=\frac{-4a^2}{Sin^2\theta} $$

**$$\therefore\,\, PQ = \vert r_1-r_2 \vert = \sqrt {(r_1+r_2)^2 – 4r_1r_2} $$ $$= 4aCosec^2\theta \ge 4a $$**

**Hence Shortest Focal Chord is Latus Rectum**

**Properties of focal Chord.**

**(1). Harmonic mean of two segments of focal chords is equal to semi latus rectum.**

Harmonic mean of PS and QS

$$H = 2(\frac{(PS)(QS)}{PS + QS }) = 2(\frac{(a(1+t_1^2))(a(1+\frac{1}{t_1^2}))}{a(t_1+\frac{1}{t_1})^2})$$

$$=\frac{ 2a^2(2+t_1^2+\frac{1}{t_1^2})}{a(t_1+\frac{1}{t_1})^2}$$

**Note : Harmonic mean of two segments of focal chord is semi latus rectum hold for all Conic Sections , including Ellipse and Hyperbola . ***Prove it post in comment form.*

**(2)**. **With focal chord as one of the diameter if a circle is ***drawn , the circle touches a fixed line i.e directrix.*

Let PQ is a diameter and $O_1$ be the mid point of PQ

$ \Rightarrow $ X – Co-ordinates of $O_1$ is $$ X_{0_1}=(\frac{at_1^2+at_2^2}{2})=\frac{a}{2}(t_1^2+t_2^2)$$

$\Rightarrow $ Distance of $X_{0_1}$ from Directrix $$ = \frac{a(t_1^2+t_2^2)}{2}+a$$ $$i.e \,\,\, X_0R=\frac{PM + QN}{2}$$

**Hence directrix is a tangent to the circle.**

**(3) . With a segment of a focal chord (i.e PS or QS ) as one of the diameter, if a circle is drawn, the circle touches the fixed-line i.e tangent at a vertex of the ***parabola.*

Let PS be a diameter of a circle with centre as $O_2\,(X_0,Y_0)$

$$X_{O_2} = \frac {(at_1^2+a)}{2} = \frac{a}{2}(1+t_1^2)$$ $$ SO = a\,\,\,\,\,\, ,\,\,\,\,\,\,PM=at_1^2$$ $$\Rightarrow \,\,\,\,\,\,X_{0_2}N=SO + PM $$

Hence tangent at vertex is a tangent to the Circle with PS as one of its diameter.

**(4). ****Tangent at **** extremities of focal chord meet each other at a point as directrix**.

Tangent to $y^2=4ax$ at any parametric coordinate P(t) is given by $$ yt=x+at^2$$

Hence Equation of tangent at P($t_1$) is PR $ yt_=x+at_1^2 $

And Equation of tangent at Q$(t_2)$ is QR , $ y_2=x+at_2^2$

Co-ordinates of point R = $(at_1t_2\,,\,a(t_1+t_2))$

Since PQ is a focal chord hence $t_1t_2=-1$

X co-ordinate of point R is $X_R=-a$

Hence point of intersection of tangent at extremities of focal chord lie as directrix.

Also Slope of tangent at $P(t_1) \,\,\,\,\,\,\,m_{PR}=\frac{1}{t_1}$

and Slope of tangent at $Q(t_2)\,\,\,\,\,\,\,m_{QR}=\frac{1}{t_2}$

$$\Rightarrow \,\,\,m_{PR}.m_{QR}=\frac{1}{t_1.t_2}=-1$$

*Tangents are perpendicular to each other.*

**Question 4 :$$ $$ Let PQ be a focal chord of a parabola with origin as a focus . Coordinates of point P and Q be (-2,0) and (4,0) respectively . Find length of latus rectum and equation of tangent at vertex of parabola.**

**Tangent and normal to the parabola**

*(1) Equation of tangent and normal to $y^2=4ax$ at a point P$(at^2,2at)$ *

Eqation of tangent $ yt=x\,+\,at^2 $

Equation of normal $y=-tx+2at+at^3$

*(2) Equation of tangent and normal to $y^2=-4ax$ at point P$(-at^2,2at)$*

Equation of tangent $yt+x=at^2$

Equation of Normal $y=tx+2at+at^3$

*(3) Equation of tangent and normal to $x^2=4ay$ at point P$(2at,at^2)$*

Equation of tangent $y=xt-at^2$

Equation of Normal $yt=-x+2at+at^3$

*(4) Equation of tangent and normal to $x^2=-4ay$ at point P$(2at,-at^2)$*

Equation of tangent $y=-xt+at^2$

Equation of Normal $yt=x-2at-at^3$

**Properties of tangent and Normal**

**(1) Length of tangent intercepted between point of contact and axis is bisected by tangent at vertex.**

P = (a$t^2$ , 2at ) , T = (-a$t^2$,0) , M = (0 , at )

Where M is the mid – point of P and T .

**(2) Length of tangent intercepted between point of contact and directrix subtend and an angle of $90^0$ at focus.**

point of intersection of tangent at P(t) and directrix is point A = (-a , a($\frac{t^2-1}{t}$))

$$\therefore \,\,\,\,\,\,\,m_{PS}=\frac{2t}{t^2-1} \,\,\,\,\,\,,\,\,\,\,\, m_{AS}=\frac{t^2-1}{-2t}$$ $$ m_{PS}.m_{AS} = -1 $$

**(3) Perpendicular from focus** **on any tangent lie on tangent at virtex of parabola**

Coordinates of perpendicular from focus on tangent at P(t) is M = ( 0 , at ) .

**(4) Image of focus about any tangent lie on the directrix.**

The foot of the perpendicular from a focus on a tangent is point M = ( 0, at )

Let Q be the image of focus S about tangent at P(t) .

$\therefore$ M as the midpoint of SQ, Hence Q( -a, 2at ).

**(5) REFLECTION PROPERTY OF PARABOLA:: Tangent and Normal at any point is the angle bisector of a line parallel to axis from point P and focal chord from point P. (i.e A ray parallel to the axis of parabola after reflection passes through the focus of parabola ). **

Since TSPQ is a rhombus with PT as QS as diagonals.

Normal PN at point P(t) is parallel to QS. Hence PT and PN are angle bisector of line PQ and PS. $$ $$ i.e any ray parallel to the axis of parabola after reflection from point of contact on parabola passes through the focus of the parabola.

**Question 5 : – $$ $$ If the tangent at the point ($X_1 , Y_1$) and ($X_2,Y_2$) on the parabola $y^2$=4ax meet at the point $(\alpha , \beta)$ and normal at the same points at (h,k) , prove that $$ (i)\,\,\, \alpha=(\frac{Y_1.Y_2}{4a})\,\,\,\,,\,\,\,\, \beta=(\frac{Y_1+Y_2}{2})$$ $$(ii)\,\,h=2a+(\frac{Y_1^2+Y_1.Y_2+Y_2^2}{4a})\,\,\,,\,\,\,k=-Y_1.Y_2(\frac{Y_1+Y_2}{8a^2})$$ $$(iii)\,\,h=2a+\frac{\beta^2}{a}-\alpha \,\,\,\,,\,\,\,\, k=-(\frac{\alpha.\beta}{a}) $$**