Parabola - Practice Questions & MCQ

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 $\mathrm{e}=1$.

$
\begin{aligned}
& \frac{P S}{P M}=e=1 \\
& \Rightarrow P S=P M
\end{aligned}
$
Standard 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}{cc} 
& \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 which passes through the focus and is perpendicular to the Directrix of the parabola. For the parabola $\mathbf{y}^2=\mathbf{4 an x}$, the X-axis is the Axis.
2. Vertex: The point of intersection of the parabola and axis. For parabola $\mathbf{y}^{\mathbf{2}}=\mathbf{4 a x}, \mathrm{A}(0,0)$ 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, $\mathrm{QQ}^{\prime}$ ' 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
$\mathrm{SP}=\mathrm{PM}=$ Distance of P from the directrix
$P=(x, y)$
$\mathrm{SP}=\mathrm{PM}=\mathrm{x}+\mathrm{a}$

Length of the Latus rectum and Parametric form

The length of the latus rectum of the parabola $y^2=4 a x {\text { is }} 4 \mathrm{a}$.

L and $\mathrm{L}^{\prime}$ be the ends of the latus rectum.
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 $\mathbf{P}(\mathbf{t})$ lying on the parabola means the coordinates of $P$ are (at ${ }^{\mathbf{2}}, \mathbf{2 a t}$ )

Other Forms of Parabola

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

Equation of the parabola is $\mathrm{y}^2=-4 \mathrm{ax}, \mathrm{a}>0$

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 $I x+m y+n=0$ be the equation of the directrix and $P(x, y)$ be any point on the parabola.

Then, from the definition PS = 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.