Parabola in Conic Sections

Parabola

• A parabola is the set of all points in a plane that are equidistant from a fixed line and a fixed point (not on the line) in the plane
• Directrix: The fixed line is called the directrix of the parabola
• Focus: the fixed-point F is called the focus
• Axis of Parabola: A line through the focus and perpendicular to the directrix is called the axis of the parabola.
• Vertex: The point of intersection of parabola with the axis is called the vertex.

Standard equations of parabola

The equation of a parabola is simplest if the vertex is at the origin and the axis of symmetry is along the x-axis or y-axis

Equation of Parabola with focus on Positive x-axis
Focus (a,0) a > 0 and directrix as x=-a

Let P (x,y) be any point on Parabola
Distance of P from Focus
$=\sqrt { (x-a)^2 + y^2}$

Distance of point P from Directrix
$=\sqrt {(x+a)^2}$
Since both of then them are equal
$\sqrt { (x-a)^2 + y^2}=\sqrt {(x+a)^2}$
$y^2=4ax$

Equation of Parabola with focus on negative x-axis
Focus (-a,0) a> 0 and directrix as x=-a

Equation is
$y^2=-4ax$

Equation of Parabola with focus on Positive y-axis
Focus (0,a) a > 0 and directrix as y=-a

Equation is
$x^2=4ay$

Equation of Parabola with focus on negative y-axis Focus(0,-a) a > 0 and directrix as y=a

Equation is
$x^2=-4ay$

Important Observation

(i)Parabola is symmetric with respect to the axis of the parabola.If the equation has a $y^2$ term, then the axis of symmetry is along the x-axis and if the equation has an $x^2$ term, then the axis of symmetry is along the y-axis.
(ii)When the axis of symmetry is along the x-axis the parabola opens to the
(a) right if the coefficient of x is positive,
(b) left if the coefficient of x is negative.
(iii)When the axis of symmetry is along the y-axis the parabola opens
(c) upwards if the coefficient of y is positive.
(d) downwards if the coefficient of y is negative

Latus Rectum Of Parabola

Latus rectum of a parabola is a line segment perpendicular to the axis of the parabola, through the focus and whose end points lie on the parabola
Latus Rectum of Parabola $y^2=4ax$ is 4a

Position of a Point with Respect to a Parabola

Let the Parabola Equation $S ≡ y^2 - 4ax = 0$
A point $(x_1,y_1)$ lies outside the parabola if $S_1 ≡ y_1^2 - 4ax_1$ is positive ($S_1 > 0$).
A point $(x_1,y_1)$ lies inside the parabola if $S_1 ≡ y_1^2 - 4ax_1$ is negative ($S_1 < 0$).
A point $(x_1,y_1)$ lies one the parabola if $S_1 ≡ y_1^2 - 4ax_1$ is Zero ($S_1 = 0$).

Parametric forms of Parabola

(i) $x=\frac {a}{m^2}$ and $y=\frac {2a}{m}$
(ii) $x=at^2$ and $y=2at$

Length of the chord intercepted by the parabola

Let Parabola $y^2 =4ax$ be intercepted by the line $y = mx + c$ is
$L=\frac {4}{m^2 } \sqrt {1+m^2} \sqrt {a(a-mc)}$

Equation of tangent of Parabola

Equation of tangent at $P(x_1,y_1)$ in a Parabola
$y^2=4ax$
Equation of tangent
$yy_1=2a(x+x_1)$
Another line equation would be
$y=mx + \frac {a}{m}$
This is at the point ($am^2,-2am$)

Equation of Normal of Parabola

Normal to any point on the Parabola is the line which passes through point and its perpendicular to the tangent at that point
Equation of Normal
$y-y_1 = \frac {- y_1}{a}(x-x_1)$

Pair of Tangents to Parabola

Two tangents, PQ and PR, can be drawn from a point $P(x_1,y_1)$ to the Parabola $S = y^2-4ax = 0$.
The combined equation is
$(y^2-4ax)(y_1^2-4ax_1)= (yy_1-2a(x+x_1))^2$
This can be written like below to remember easily
$SS_1 = T^2$,
Where S = 0 is the equation of the circle, T = 0 is the equation of tangent at $(x_1, y_1)$, and S1 is obtained by replacing x by $x_1$ and y by $y_1$ in S.

Chord of Contact of Parabola

The chord joining the two points of contact of tangents to a circle drawn from any point P is called the chord of contact of P with respect to the given circle.
Equation of Chord of Contact
$yy_1-2a(x+x_1)=0$

Special Theorem

Theorem I
The circle circumscribing the triangle formed by any three tangents to a parabola passes through the focus.
Theorem II
The orthocenter of any triangle formed by three tangents to a parabola $y^2 = 4ax$ lies on then directrix and has the coordinates $–a, a(t_1 + t_2 + t_3 + t_1t_2t_3)$.
Theorem III
The area of the triangle formed by three points on a parabola is twice the area of the triangle formed by the tangents at these points.