If the focus is $(\alpha ,\beta )$ and the directrix is $ax+by+c=0$, then the equation of the conic section

whose eccentricity $=e$ is $SP=ePM$


$\Rightarrow \sqrt{(x-\alpha {)}^2+(y-\beta {)}^2}=e\cdot \frac{|ax+by+c|}{\sqrt{\left(a^2+b^2\right)}}$

$\Rightarrow (x-\alpha {)}^2+(y-\beta {)}^2=e^2\cdot \frac{(ax+by+c{)}^2}{\left(a^2+b^2\right)}$.

Consider the focus of the parabola as $S(a,0)$ and directrix be $x+a=0$, and axis as $x$-axis.


Now according to the definition of the parabola, for any point on the parabola, we must have

$SP=PM$

$\Rightarrow \sqrt{(x-a{)}^2+(y-0{)}^2}=PN+NM=x+a$

$\Rightarrow (x-a{)}^2+y^2=(a+x{)}^2$

$\Rightarrow y^2=(a+x{)}^2-(x-a{)}^2$

$\Rightarrow y^2=4ax$

Vertex : $(0,0)$

Tangent at vertex : $x=0$

Equation of latus rectum : $x=a$

Extremities of latus rectum : $(a,2a),(a,-2a)$

Length of latus rectum : $4a$

Focal distance (SP) : $SP=PM=x+a$

Parametric form : $x=a{t}^2$ and $y=2at$, where $t$ is parameter

(1) Parabola opening to left (i.e. $y^2=-4ax):(a>0)$


(i) Vertex is $A(0,0)$.

(ii) Focus is $S(-a,0)$.

(iii) Equation of the directrix $MZ$ is $x-a=0$.

(iv) Equation of the axis is $y=0$ i.e. $X$-axis.

(v) Equation of the tangent at the vertex is $x=0$ i.e. Y-axis.

(vi) Length of latusrectum $=L{L}^{\prime }=4a$.

(vii) Ends of latusrectum are $L(-a,2a)$ and $L^{\prime }(-a,-2a)$.

(viii) Equation of latusrectum is $x=-a$ i.e. $x+a=0$.

(ix) Parametric coordinates is $\left(-a{t}^2,2at\right)$.

(2) Parabola opening upwards (i.e. $x^2=4ay):(a>0)$


(i) Vertex is $A(0,0)$.

(ii) Focus is $S(0,a)$.

(iii) Equation of the directrix $MZ$ is $y+a=0$.

(iv) Equations of the axis is $x=0$ i.e. $Y$-axis.

(v) Equation of the tangent at the vertex is $y=0$ i.e. $X$-axis.

(vi) Ends of latusrectum are $L(2a,a)$ and $L^{\prime }(-2a,a)$.

(vii) Length of latusrectum $=L{L}^{\prime }=4a$.

(viii) Equation of latusrectum is $y=a$ i.e. $y-a=0$.

(ix) Parametric coordinates is $\left(2at,a{t}^2\right)$.

(3) Parabola opening downwards (i.e. $x^2=-4ay):(a>0)$


(i) Vertex is $A(0,0)$. (ii) Focus is $S(0,-a)$.

(iii) Equation of the directrix $MZ$ is $y-a=0$.

(iv) Equation of the axis is $x=0$ i.e. $Y$-axis.

(v) Equation of the tangent at the vertex is $y=0$ i.e. $X$-axis.

(vi) Length of latusrectum $=L{L}^{\prime }=4a$

(vii) Ends of latusrectum are $L(2a,-a)$ and $L^{\prime }(-2a,-a)$

(viii) Equation of latusrectum is $y=-a$ i.e. $y+a=0$.

(ix) Parametric coordinates are $\left(2at,-a{t}^2\right)$.

Let $S(a,b)$ be the focus, and $lx+my+n=0$ is the equation of the directrix. Let $P(x,y)$ be any point on the parabola.

Then by definition $SP=PM$



$\Rightarrow \sqrt{(x-a{)}^2+(y-b{)}^2}=\frac{|lx+my+n|}{\sqrt{\left(l^2+m^2\right)}}$

$\Rightarrow (x-a{)}^2+(y-b{)}^2=\frac{(lx+my+n{)}^2}{\left(l^2+m^2\right)}$

$\Rightarrow m^2{x}^2+l^2{y}^2-2lmxy+x$ term $+y$ term $+$ constant $=0$

This is of the form $(mx-ly{)}^2+2gx+2fy+c=0$.

This equation is the general equation of parabola.

The parabola

$$ y^2=4 a x $$

can be written as $(y-0)^2=4 a(x-0)$.

The vertex of this parabola is $A(0,0)$



Now, when origin is shifted at $A^{\prime}(h, k)$ without changing the direction of axes, its equation becomes

$$ (y-k)^2=4 a(x-h) $$

Note : 1. Another form is $(x-h)^2=4 a(y-k)$ axis parallel to $Y$-axis with its vertex $(h, k)$ its focus is at $(h, a+k)$ and length of latusrectum $=4 a$, the equation of the directrix is

$$ y=k-a \Rightarrow y+a-k=0 . $$

2. The parametric equation of $(y-k)^2=4 a(x-h)$ are $x=h+a t^2$ and $y=k+2 a t$.

The point $\left(x_1,y_1\right)$ lies outside, on or inside the parabola $y^2=4ax$ according as

$y_1^2-4a{x}_1>,=\text{, or }<0$

Let $y^2=4ax$ be a parabola, if $PQ$ be a focal chord.

Then, $P\equiv \left(a{t}_1^2,2a{t}_1\right)$ and $Q\equiv \left(a{t}_2^2,2a{t}_2\right)$

Since, $PQ$ passes through the focus $S(a,0)$.

$\therefore Q,S,P$ are collinear.


$\therefore$ Slope of $PS=$ Slope of $QS$

$\Rightarrow \frac{2a{t}_1-0}{a{t}_1^2-a}=\frac{0-2a{t}_2}{a-a{t}_2^2}\Rightarrow \frac{2t_1}{t_1^2-1}=\frac{2t_2}{t_2^2-1}$

$\Rightarrow t_1\left(t_2^2-1\right)=t_2\left(t_1^2-1\right)$

$\Rightarrow t_1{t}_2\left(t_2-t_1\right)+\left(t_2-t_1\right)=0$

$\Rightarrow t_2-t_1\ne 0$ or $t_1{t}_2+1=0$

$\Rightarrow t_1{t}_2=-1$ or $t_2=-\frac{1}{t_1}$,

Note :

If one extremity of a focal chord is $\left(a{t}_1^2,2a{t}_1\right)$ then the other extremity $\left(a{t}_2^2,2a{t}_2\right)$ becomes $\left(\frac{a}{t_1^2},-\frac{2a}{t_1}\right)$.

The equation of a chord of the parabola $y^2=4ax$ joining its two points $P\left(t_1\right)$ and $Q\left(t_2\right)$ is $y\left(t_1+t_2\right)=2x+2t_1{t}_2$

Note :

(i) If $PQ$ is focal chord then $t_1{t}_2=-1$.

(ii) Extremities of focal chord can be taken as $\left(a{t}^2,2at\right)\&\left(\frac{a}{t^2},\frac{-2a}{t}\right)$

(iii) If ${\mathrm{t}}_1{\mathrm{t}}_2=\mathrm{k}$ then chord always passes a fixed point $(-\mathrm{ka},0)$.

(a) The line $y=mx+c$ meets the parabola $y^2=4ax$ in two points real, coincident or imaginary according as a > = < cm.

$\Rightarrow$ condition of tangency is, $c=\frac{a}{m}$.

Note : Line $y=mx+c$ will be tangent to parabola

${\mathrm{x}}^2=4$ay if $\mathbf{c}=-{\mathbf{am}}^2$.

(b) Length of the chord intercepted by the parabola $y^2=4$ ax on the line $y=mx+c$ is :

$\left(\frac{4}{m^2}\right)\sqrt{a\left(1+m^2\right)(a-mc)}$.

Note : length of the focal chord making an angle $\alpha$ with the $x$-axis is $4a{\mathrm{cosec}}^2\alpha$.

PT and PG are the tangent and normal respectively at the point $\mathrm{P}$ to the parabola $y^2=4ax$.

Then $\mathrm{TN}=$ length of subtangent $=$ twice the abscissa of the point $\mathrm{P}$ (Subtangent is always bisected by the vertex)

$NG=$ length of subnormal which is constant for all points on the parabola and equal to its semi latus rectum (2a).

(a) Point form :

Equation of tangent to the given parabola at its point $\left(x_1,y_1\right)$ is $y{y}_1=2a\left(x+x_1\right)$

(b) Slope form :

Equation of tangent to the given parabola whose slope is ' $m$ ', is $y=mx+\frac{a}{m},(m\ne 0)$

Point of contact is $\left(\frac{\mathrm{a}}{{\mathrm{m}}^2},\frac{2\mathrm{a}}{\mathrm{m}}\right)$

(c) Parametric form :

Equation of tangent to the given parabola at its point $\mathrm{P}(\mathrm{t})$, is $\mathrm{ty}=\mathrm{x}+\mathrm{at}{}^2$

Note : Point of intersection of the tangents at the point ${\mathrm{t}}_1\&{\mathrm{t}}_2$ is $[a{t}_1{t}_2,a\left(t_1+t_2\right]$ ]. (i.e. G.M. and A.M. of abscissae and ordinates of the points)

(a) Point form : Equation of normal to the given parabola at its point

$\left(x_1,y_1\right)$ is $y-y_1=-\frac{y_1}{2a}\left(x-x_1\right)$

(b) Slope form : Equation of normal to the given parabola whose slope is ' $m$ ', is

$y=mx-2am-a^3$ foot of the normal is $\left(a^2,-2am\right)$

(c) Parametric form :

Equation of normal to the given parabola at its point $\mathrm{P}(\mathrm{t})$, is $y+tx=2at+a{t}^3$

Note :

(i) Point of intersection of normals at ${\mathrm{t}}_1\&{\mathrm{t}}_2$ is $\left(a\left(t_1^2+t_2^2+t_1{t}_2+2\right),-a{t}_1{t}_2\left(t_1+t_2\right)\right)$.

(ii) If the normal to the parabola ${\mathrm{y}}^2=4\mathrm{ax}$ at the point ${\mathrm{t}}_1$, meets the parabola again at the point ${\mathrm{t}}_2$, then $t_2=-\left(t_1+\frac{2}{t_1}\right)$.

(iii) If the normals to the parabola ${\mathrm{y}}^2=4\mathrm{ax}$ at the points ${\mathrm{t}}_1\&$ ${\mathrm{t}}_2$ intersect again on the parabola at the point ' ${\mathrm{t}}_3$ ' then ${\mathrm{t}}_1{\mathrm{t}}_2=2;{\mathrm{t}}_3=-\left({\mathrm{t}}_1+{\mathrm{t}}_2\right)$ and the line joining ${\mathrm{t}}_1\&{\mathrm{t}}_2$ passes through a fixed point $(-2\mathrm{a},0)$.

The equation of the pair of tangents which can be drawn from any point $\mathrm{P}\left({\mathrm{x}}_1,{\mathrm{y}}_1\right)$ outside the parabola to the parabola ${\mathrm{y}}^2=4\mathrm{ax}$ is given by : ${\mathrm{SS}}_1={\mathrm{T}}^2$, where :

$S\equiv y^2-4ax;S_1\equiv y_1^2-4a{x}_1;T\equiv y{y}_1-2a\left(x+x_1\right).$

Equation of the chord of contact of tangents drawn from a point $\mathrm{P}\left({\mathrm{x}}_1,{\mathrm{y}}_1\right)$ is $y{y}_1=2\mathrm{a}\left(\mathrm{x}+{\mathrm{x}}_1\right)$

Remember that the area of the triangle formed by the tangents from the point $\left({\mathrm{x}}_1,{\mathrm{y}}_1\right)$ and the chord of contact is $\frac{{\left({\mathrm{y}}_1^2-4{\mathrm{ax}}_1\right)}^{3/2}}{2\mathrm{a}}$. Also note that the chord of contact exists only if the point $\mathrm{P}$ is not inside.

Equation of the chord of the parabola $\mathrm{y}^2=4 \mathrm{ax}$ whose middle point is

$\left(x_1, y_1\right)$ is $y-y_1=\frac{2 a}{y_1}\left(x-x_1\right)$.

This reduced to $T=S_1$

where $T \equiv y y_1-2 a\left(x+x_1\right) \quad \& \quad S_1 \equiv y_1{ }^2-4 a x_1$.

The locus of the middle points of a system of parallel chords of a Parabola is called a DIAMETER. Equation to the diameter of a parabola $y^2=4ax$ is $y=2a/m$, where $m=$ slope of parallel chords.