Ellipse - Practice Questions & MCQ

Ellipse

An ellipse is the set of all points $(x, y)$ in a plane such that the sum of their distances from two fixed points is a constant. Each fixed point is called a focus (plural: foci).

OR

The locus of a point which moves in a plane such that the ratio of the distance from a fixed point (focus) to the distance from a fixed line (directrix) is constant. Constant is known as eccentricity e and for ellipse $\mathrm{e}<1$.

Consider the figure, C is the origin, S is the focus and ZM is the directrix.
$S A=e \cdot A Z$
(i)
$S A^{\prime}=e \cdot A^{\prime} Z$
(from the definition)
Adding these
Let $A A^{\prime}=2 a . C$ is the mid point of $A A^{\prime}$. Hence, $C A=C A^{\prime}=\mathrm{a}$.
$A=(a, 0)$ and $A^{\prime}=(-a, 0)$
From equation (i) and (ii)
$S A+S A^{\prime}=e\left(A Z+A^{\prime} Z\right)$
$2 \mathrm{a}=\mathrm{e}\left(C Z-C A+C A^{\prime}+C Z\right)$
$2 \mathrm{a}=\mathrm{e}(2 \mathrm{CZ}) \quad\left[\mathrm{CA}=\mathrm{CA}^{\prime}\right]$
$\mathrm{CZ}=\mathrm{a} / \mathrm{e}$
The equation of directrix, ZM is $\mathrm{x}=\mathrm{a} / \mathrm{e}$
Again from equation (i) and (ii)
$S A^{\prime}-S A=e\left(A^{\prime} Z-A Z\right)$
$\left[\left(C A^{\prime}+C S\right)-(C A-C S)\right]=e\left[A A^{\prime}\right]$
$2 C S=e(2 a)$
$\mathrm{CS}=\mathrm{ae}$
The coordinate of focus, $S=(a e, 0)$ and $S^{\prime}=(-a e, 0)$

P(x, y) is any point on the ellipse and PM is perpendicular to directrix ZM.

$\begin{aligned} & \frac{\mathrm{SP}}{\mathrm{PM}}=\mathrm{e} \Rightarrow(\mathrm{SP})^2=\mathrm{e}^2(\mathrm{PM})^2 \\ & (\mathrm{x}-\mathrm{ae})^2+(\mathrm{y}-0)^2=\mathrm{e}^2\left(\frac{\mathrm{a}}{\mathrm{e}}-\mathrm{x}\right)^2 \\ & \mathrm{x}^2+\mathrm{a}^2 \mathrm{e}^2-2 \mathrm{aex}+\mathrm{y}^2=\mathrm{e}^2 \mathrm{x}^2-2 \mathrm{aex}+\mathrm{a}^2 \\ & \mathrm{x}^2\left(1-\mathrm{e}^2\right)+\mathrm{y}^2=\mathrm{a}^2\left(1-\mathrm{e}^2\right) \\ & \frac{\mathrm{x}^2}{\mathrm{a}^2}+\frac{\mathrm{y}^2}{\mathrm{a}^2\left(1-\mathrm{e}^2\right)}=1 \\ & \frac{\mathrm{x}^2}{\mathrm{a}^2}+\frac{\mathrm{y}^2}{\mathrm{~b}^2}=1, \quad \mathrm{~b}^2=\mathrm{a}^2\left(1-\mathrm{e}^2\right)\end{aligned}$

Important Terms related to ellipse:

Centre: All chord passing through point $C$ is bisected at point $C$. Here $C$ is the origin, i.e. $(0,0)$.
Foci: Point $S$ and $S^{\prime}$ is foci of the ellipse where, $S$ is $(a e, 0)$ and $S^{\prime}$ is $(-a e, 0)$.
Directrices: The straight-line ZM and Z'M' are two directrices of the ellipse and their equations are $x=a / e$ and $x=-a / e$.
Axis: In figure AA' is called the major axis and BB' is called the minor axis. $2 a$ is called the length of the major axis and $2 b$ is called the length of the minor axis.

Double Ordinate: If a line perpendicular to the major axis meets the curve at P and $\mathrm{P}^{\prime}$, then PP ' is called double ordinate.
Latusrectum: Double ordinate passing through focus is called latus rectum. Here LL' is a latus rectum. There is another latus rectum that passes through the other focus $\mathrm{S}^{\prime}$. So an ellipse has 2 latus rectum

Standard Equation of Ellipse:

The standard form of the equation of an ellipse with center (0, 0) and major axis on the x-axis is

$\frac{\mathbf{x}^2}{\mathbf{a}^2}+\frac{\mathbf{y}^2}{\mathbf{b}^2}=1 \quad$ where, $b^2=a^2\left(1-e^2\right)$

$
a>b
$

the length of the major axis is 2 a the length of the minor axis is $2 b$ the coordinates of the vertices are $( \pm a, 0)$