Parabola - General Equations, Properties and Practice Problems

The four curves - circle, parabola, ellipse, and hyperbola are called conic sections because they can be formed by interesting a double right circular cone with a plane. A parabola is the locus of a point moving in a plane such that its distance from a fixed point (focus) is equal to its distance from a fixed line (directrix). In real life, we use Parabolas in bridges, telescopes, satellites, etc.

In this article, we will cover the concept of Parabola. This category falls under the broader category of Coordinate Geometry, which is a crucial Chapter in class 11 Mathematics. It is not only essential for board exams but also for competitive exams like the Joint Entrance Examination(JEE Main) and other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE, and more. A total of Eighteen questions have been asked on JEE MAINS( 2013 to 2023) from this topic including three in 2020, four in 2021, four in 2022, and one in 2023.

What is Parabola?

A parabola is the locus of a point moving in a plane such that its distance from a fixed point (focus) is equal to its distance from a fixed line (directrix).

Hence it is a conic section with eccentricity e = 1.

$\begin{aligned} & \frac{P S}{P M}=e=1 \\ & \Rightarrow P S=P M\end{aligned}$

Standard equation of a parabola

If the directrix is parallel to the y-axis in the standard equation of a parabola is given as
$
y^2=4 a x
$
If the directrix is parallel to the $x$-axis, the standard equation of a parabola is given as

$
x^2=4 a y
$

Derivation of equation of a parabola

Let focus of parabola is $S(a, 0)$ and directrix be $x+$ $a=0$
$P(x, y)$ is any point on the parabola.
Now, from the definition of the parabola,

$
\begin{array}{lc}
& \mathrm{SP}=\mathrm{PM} \\
\Rightarrow & \mathrm{SP}^2=\mathrm{PM}^2 \\
\Rightarrow & (\mathrm{x}-\mathrm{a})^2+(\mathrm{y}-0)^2=(\mathrm{x}+\mathrm{a})^2 \\
\Rightarrow & \mathrm{y}^2=4 \mathrm{ax}
\end{array}
$

which is the required equation of a standard parabola

Important Terms related to Parabola

1. Axis: The line that passes through the focus and is perpendicular to the Directrix of the parabola. For parabola $y^2=4 \mathrm{ax}_{\text {, }}$ the x -axis is the Axis.
2. Vertex: The point of intersection of the parabola and axis. For parabola $y^2=4 \mathbf{a x}$, $A(0,0)$ i.e. origin is the Vertex.
3. Double Ordinate: Suppose a line perpendicular to the axis of the parabola meets the curve at Q and $\mathrm{Q}^{\prime}$. Then, QQ ' is called the double ordinate of the parabola.
4. Latus Rectum: The double ordinate LL' passing through the focus is called the latus rectum of the parabola.
5. Focal Chord: A chord of a parabola which is passing through the focus. In the figure PP' and LL' are the focal chord.
6. Focal Distance: The distance from the focus to any point on the parabola. I.e. PS

$
\begin{aligned}
& \mathrm{SP}=\mathrm{PM}=\text { Distance of } \mathrm{P} \text { from the directrix } \\
& \mathrm{P}=(x, y) \\
& \mathrm{SP}=\mathrm{PM}=\mathrm{x}+\mathrm{a}
\end{aligned}
$

Length of the Latus rectum

The length of the latus rectum of the parabola $y^2=4 ax$ is 4$a$.

Derivation of Length of the Latus Rectum

$L$ and $L^{\prime}$ be the ends of the latus rectum.
Clearly the x-coordinate of $L$ and $L$ ' will be 'a'
As $L$ and $L^{\prime}$ lie on parabola, so put $x=a$ in the equation of parabola to get $y$ coordinates of $L$ and $L^{\prime}$

$
\begin{aligned}
& y^2=4 a \cdot a=4 a^2 \\
& \Rightarrow y= \pm 2 a \\
& \therefore L(a, 2 a) \text { and } L^{\prime}(a,-2 a) \\
& \text { so that } L L^{\prime}=4 a
\end{aligned}
$

Parametric Equation

From the equation of the parabola, we can write $\frac{y}{2 a}=\frac{2 x}{y}=t$ here, t is a parameter
Then, $x=a t^2$ and $y=2 a t$ are called the parametric equations and the point $\left(a t^2, 2 a t\right)$ lies on the parabola.

Point $\mathrm{P}(\mathrm{t})$ lying on the parabola means the coordinates of P are (at $\left.{ }^2, 2 a t\right)$

Other Forms of Parabola

1. Parabola with focus $S(-a, 0)$ and directrix $x=a$ (Parabola opening to left)

Equation of the parabola is $y^2=-4 a x, a>0$

2. Parabola with focus $S(0, a)$ and directrix $y=-a$ (Parabola opening to upward)

The equation of the parabola is $x^2=4 a y, a>0$.

3. Parabola with focus $S(0,-a)$ and directrix $y=a$ (Parabola opening to downward)

The equation of the parabola is $x^2=-4 a y, a>0$.

General equation of Parabola

Let $S(h, k)$ be the focus and $\mathrm{l} x+m y+n=0$ be the equation of the directrix, and $\mathrm{P}(\mathrm{x}, \mathrm{y})$ be any point on the parabola.

Then, from the definition PS $=\mathrm{PM}$

$
\Rightarrow \quad \sqrt{(x-h)^2+(y-k)^2}=\left|\frac{l x+m y+n}{\sqrt{\left(l^2+m^2\right)}}\right|
$
Squaring both sides, we get

$
\Rightarrow \quad(x-h)^2+(y-k)^2=\frac{(l x+m y+n)^2}{\left(l^2+m^2\right)}
$
This is the general equation of a parabola.

Focal Chord and Focal Distance

Focal chord: Any chord that passes through the focus of the parabola is the focal chord of the parabola. It cuts the parabola at two distinct points.

In the figure S is the focus and PQ is the focal chord.

Focal Distance: The distance of any point, p(x, y) on the parabola from the focus, is the focal distance. PS is the focal distance in the above figure.

Position of point with respect to Parabola

To get the point(s) of intersection, let us solve the equations of the parabola and the line simultaneously

Parabola is $\mathrm{y}^2=4 \mathrm{ax}$ and a line $\mathrm{y}=\mathrm{mx}+\mathrm{c}$
then, $y^2=4 a\left(\frac{y-c}{m}\right)$

$
\Rightarrow m y^2-4 a y+4 a c=0
$
The above equation is quadratic in $y$
Depending on the discriminant of this equation, if we have 2 real roots, then 2 distinct intersection points and line is a chord/secant If we have 2 equal roots, then we have only one point where line touches the parabola and the line is a tangent
If we do not have any real roots, then line does not intersect the parabola

Properties of Parabola
1. The portion of a tangent to a parabola intercepted between the directrix and the curve subtends a right angle at the focus.

2. The tangent at a point P on the parabola y² = 4ax is the bisector of the angle between the focal radius and the perpendicular from P on the directrix.

3. The foot of the perpendicular from the focus on any tangent to a parabola lies on the tangent at the vertex.

4. If S is the focus of the parabola and tangent and normal at any point P meets its axis in T and G respectively, then ST = SG = SP

Chord of Contact

S is a parabola and P(x₁,y₁) is an external point to parabola S. A and B are the points of contact of the tangents drawn from P to parabola S. Then the chord AB is called the chord of contact of the parabola S drawn from an external point P.

The equation of the chord of the parabola S=y²-4ax=0 , from an external point P(x₁,y₁) is
$\mathrm{T}=0$ or $\mathbf{y} \mathbf{y}_1-2 \mathrm{a}\left(\mathrm{x}+\mathrm{x}_1\right)=0$

Important Properties of Focal Chord

1. If chord joining P = (at₁², 2at₁) and Q = (at₂², 2at₂) is focal chord of parabola y² = 4ax, then t₁t₂ = -1.
2. If one extremity of a focal chord is (at₁², 2at₁), then the other extremity (at₂², 2at₂) becomes (a/t₁² , -2a/t₁).
3. If the point (at², 2at) lies on parabola y² = 4ax, then the length of focal chord PQ is a(t + 1/t)².
4. The length of the focal chord, which makes an angle θ with a positive x-axis, is 4a cosec² θ.
5. Semi latus rectum is a harmonic mean between the segments of any focal chord.
6. Circle described on focal length as diameter touches tangent at the vertex. The circle, described on any focal chord of a parabola as diameter, touches the directrix.

Four Common Forms of a Parabola

Solved Examples Based on Parabolas

Example 1: Three urns A, B, and C contain 4 red, 6 black; 5 red, 5 black; and $\lambda$ red, 4 black balls respectively. One of the urns is selected at random and a ball is drawn. If the ball drawn is red and the probability that it is drawn from urn C is 0.4 then the square of the length of the side of the largest equilateral triangle, inscribed in the parabola $y^2=\lambda x \space {\text {with one vertex }}$ at the vertex of the parabola, is
[JEE MAINS 2023]

Solution

$\begin{aligned} & P(\text { Red from } C)=\frac{\frac{1}{3} \times \frac{\lambda}{\lambda+4}}{\frac{1}{3} \cdot \frac{\lambda}{\lambda+4}+\frac{1}{3} \cdot \frac{4}{10}+\frac{1}{3} \cdot \frac{5}{10}} \\ & =\frac{\frac{\lambda}{\lambda+4}}{\frac{\lambda}{\lambda+4}+\frac{9}{10}} \\ & \Rightarrow \frac{10 \lambda}{10 \lambda+9(\lambda+4)}=\frac{4}{10} \\ & \Rightarrow 100 \lambda=40 \lambda+36 \lambda+144 \\ & 24 \lambda=144 \\ & \lambda=6\end{aligned}$

$\begin{gathered}\mathrm{m}=\frac{2}{t}=\frac{1}{\sqrt{3}} \\ t=2 \sqrt{3} \\ P(12 \mathrm{a}, 4 \sqrt{3} a) \\ (\text { Side })^2=144 a^2+48 a^2 \\ =192 \times \frac{9}{4}=432\end{gathered}$

Hence, the answer is 432.

Example 2: Let a common tangent to the curves $y^2 2=4 x$ and $(x-4)^2+y^2=16$ touch the curves at the points P and Q . Then $(P Q)^2$ is equal to $\qquad$
[JEE MAINS 2023]

Solution

$
\begin{aligned}
& y^2=4 x \\
& (x-4)^2+y^2=16
\end{aligned}
$

Let equation of tangent of parabola

$
y=m x+1 / m
$
Now equation 1 also touches the circle

$
\begin{aligned}
& \therefore\left|\frac{4 m+1 / m}{\sqrt{1+m^2}}\right|=4 \\
& \left(4 \mathrm{~m}^2+1 / \mathrm{m}\right)^2=16+16 \mathrm{~m}^2 \\
& 16 \mathrm{~m}^4+8 \mathrm{~m}^2+1=16 \mathrm{~m}^2+16 \mathrm{~m}^4 \\
& 8 \mathrm{~m}^2=1 \\
& \mathrm{~m}^2=1 / 8 \quad\left\{m^4=0\right\}(\mathrm{m} \rightarrow \infty)
\end{aligned}
$
For distinct points consider only $m^2=1 / 8$.
Point of contact of the parabola

$
\begin{aligned}
& P(8,4 \sqrt{2}) \\
& \therefore P Q=\sqrt{S_1} \Rightarrow(P Q)^2=S_1 \\
& =16+32-16=32
\end{aligned}
$

Hence, the answer is 32.

Example 3: Let $\mathrm{P}_1$ be a parabola with vertex $(3,2)$ and focus $(4,4)$ and $P_2$ be its mirror image with respect to the line $x+2 y=6$. Then the directrix of $P_2$ is $x+2 y=$ $\qquad$
[JEE MAINS 2022]
Solution
For $P_1$ :
slope of axis $=\frac{4-2}{4-3}=2$
Distance between vertex and focus: $\sqrt{5}$
Now line is perpendicular to axis
focus of $P_2: \frac{x-4}{1}=\frac{y-4}{2}=\frac{-2(4+8-6)}{5}$

$
\begin{aligned}
& \mathrm{x}=\frac{-12}{5}+4=\frac{8}{5} \\
& \mathrm{y}=\frac{-24}{5}+4=\frac{-4}{5}
\end{aligned}
$
Let directrix of $\mathrm{P}_2: x+2 y=\lambda$

$
\begin{aligned}
& \therefore\left|\frac{\frac{8}{5}-\frac{8}{5}-\lambda}{\sqrt{5}}\right|=2 \sqrt{5} \\
& |-\lambda|=10 \\
& \lambda= \pm 10
\end{aligned}
$

$\lambda=-10$ is rejected because directrix can't cut parabola

$
\therefore \lambda=10
$

Hence, the answer is 10.

Example 4: If the length of the latus rectum of a parabola, whose focus is $(a, a)$ and the tangent at its vertex is $\mathrm{x}+\mathrm{y}=\mathrm{a}$, is 16, then $|\mathrm{a}|$ is equal to
[JEE MAINS 2022]
Solution
Distance between focus and tangent at vector

$
\begin{aligned}
& =\frac{|a+a-a|}{\sqrt{1+1}} \\
& =\frac{|a|}{\sqrt{2}}
\end{aligned}
$
Length of $LR=\frac{4|a|}{\sqrt{2}}=16$

$
\Rightarrow|\mathrm{a}|=4 \sqrt{2}
$

$\therefore$ option $(\mathrm{c})$
Hence, the answer is $4 \sqrt{2}$

Summary

A parabola is a curve that is known for its simple but versatile equation. Its different properties make it important in theoretical as well as practical life. Understanding parabolas helps us in our daily lives such as describing the path of balls, the arc of bridges, etc.