An **ellipse** is the locus of a point in a plane such that the sum of the distances of the point from two fixed points is constant.

The two fixed points $S$ and $S’$ are the **foci** of the ellipse and the midpoint between them is the **centre** of the ellipse. In the figure given above, we take the origin $O$ as the centre of the ellipse.

The line through the two foci is known as the **major axis** of the ellipse. And the line perpendicular to the major axis is known as its **minor axis**. The intersection of the ellipse and the major axis gives two points $A$ and $A’$ which are called **vertices**.

An ellipse can also be defined as the locus of a point in a plane such that the ratio (called **eccentricity**) of its distance from a fixed point (called the **focus**) to its distance from a fixed straight line (called the **directrix**) is constant. The eccentricity (denoted by $e$) is any number between $0$ and $1$.

## Standard Equation

Let the centre of the ellipse be at the origin $O$ and the major axis along the x-axis. We call this the ellipse in the **standard position**. Let the two foci be $S(c,0)$ and $S(-c,0)$ where $c$ is a positive constant.

Take any point $P(x,y)$ on the ellipse, then \[PS+PS’=\text{Constant}\] \[\text{Let}\;PS+PS’=2a\] \[\text{Obviously,}\;2s>2a\Rightarrow a>c\] Using distance formula, \[\sqrt{(x+c)^2+y^2}+\sqrt{(x-c)^2+y^2}=2a\] \[(x+c)^2+y^2=4a^2-4a\sqrt{(x-c)^2+y^2}+(x-c)^2+y^2\] \[4cx-4a^2=-4a\sqrt{(x-c)^2+y^2}\] \[(cx-a^2)^2=a^2[(x-c)^2+y^2]\] \[c^2x^2-2a^2cx+a^4=a^2[x^2-2cx+c^2+y^2]\] \[a^2(a^2-c^2)=(a^2-c^2)x^2+a^2y^2\] \[\therefore\frac{x^2}{a^2}+\frac{y^2}{a^2-c^2}=1\]

$a^2-c^2$ is a positive constant. Let $b^2=a^2-c^2$. Then the equation of the ellipse becomes \[\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\] The vertices of the ellipse are $A(a,0)$ and $A'(-a,0)$. The distance between them, $2a$, is called the **length of the major axis**. The ellipse intersects the y-axis at points $B(0,b)$ and $B(0,-b)$. The distance between them, $2b$, is called the **length of the minor axis**.

Let $ZM$ be the directrix of the ellipse. Then, the distance of a point on the ellipse from the focus and the distance of that point from the directrix bears a constant ratio. This constant ratio is called **eccentricity** denoted by $e$. Therefore \[e=\frac{AS}{AZ}\;\text{and}\;e=\frac{A’S}{A’Z}\] \[\therefore A’S-AS=e(A’Z-AZ)\] \[(A’O+OS)-(OA-OS)=e\,AA’\] \[2OS=2ae\] \[\therefore OS=ae\;\text{i.e.}\;c=ae\]

Hence, we have \[e=\frac{c}{a}=\frac{\sqrt{a^2-b^2}}{a}=\sqrt{1-\frac{b^2}{a^2}}\] Since $c<a$, $e$ is always between $0$ and $1$. The coordinates of the foci are $(\pm ae,0)$. Also, \[AS+A’S=e(AZ+A’Z)\] \[2a=2eOZ\] \[\therefore OZ=\frac{a}{e}\] Hence, the equation of the directrix is $x=\pm\frac{a}{e}$.

If in the equation of an ellipse, \[\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\] $b>a>0$, then the major axis will be along the y-axis and the minor axis along the x-axis. In this case, $b^2=a^2+c^2$ and \[e=\frac{c}{b}=\frac{\sqrt{b^2-a^2}}{b}=\sqrt{1-\frac{a^2}{b^2}}\] Also, the equation of the directrix will be $y=\pm\frac{b}{e}$.

## Equation of an Ellipse whose Centre is not at the Origin [Centre at $(h,k)$]

Let the centre of the ellipse be at $(h,k)$. Then the equation of the ellipse is given by \[\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1\;(a>b>0)\]

The major axis and minor axis are parallel to x-axis and y-axis respectively. The vertex of the ellipse is $(h\pm a,k)$ and focus is $(h\pm ae,k)$. Eccentricity $(e)$ is given by \[e=\sqrt{1-\frac{b^2}{a^2}}\] And, the equation of the directrix is $x=h\pm\frac{a}{e}$.

Similarly, If in the equation \[\frac{(x-h)^2}{a^2}+\frac{(y-h)^2}{b^2}=1\] $b>a>0$, then the major axis and minor axis will be parallel to y-axis and x-axis respectively. The vertex will be $(h,k\pm b)$ and focus will be $(h,k\pm be)$. Eccentricity $(e)$ is given by \[e=\sqrt{1-\frac{a^2}{b^2}}\] And, the equation of the directrix is $y=k\pm\frac{b}{e}$.

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