PARABOLA

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

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 $$ (i)\,\,\, \alpha=(\frac{Y_1.Y_2}{4a})\,\,\,\,,\,\,\,\, \beta=(\frac{Y_1+Y_2}{2})$$ $$(ii)\,\,h=2a+(\frac{Y_1^2+Y_1.Y_2+Y_2^2}{4a})\,\,\,,\,\,\,k=-Y_1.Y_2(\frac{Y_1+Y_2}{8a^2})$$ $$(iii)\,\,h=2a+\frac{\beta^2}{a}-\alpha \,\,\,\,,\,\,\,\, k=-(\frac{\alpha.\beta}{a}) $$