Ellipse
2.0 Standard equation of Ellipse
2.0 Standard equation of Ellipse
$$\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$$
where ${b^2} = {a^2}(1 - {e^2})$
Let $S$ be the focus and $ZM$ the directrix of the ellipse. Draw $SZ$ perpendicular to $ZM$ and divide $SZ$ internally and externally in the ratio $e:1(e < 1)$ and let $A$ and $A'$ be these internal and external point of division.
then $$\frac{{SA}}{{AZ}} = \frac{e}{1}$$ or $$SA = e \times AZ...(1)$$
Similarly, $$S{A'} = e \times {A'}Z...(2)$$
Let $AA'=2a$ and $C$ be the mid-point of $AA'$ as origin.
Therefore, $CA=CA'=a...(3)$
Let $P(x,y)$ be any point on the ellipse. On adding equations $(1)$ and $(2)$, we get
$$SA+SA'=e(AZ+A'Z)$$$$AA'=e(CZ-CA+CA'+CZ)$$$$2a=e(2CZ)$$$$CZ=a/e$$
The equation of directrix $MZ$ is $x = \frac{a}{e}$ $$\frac{a}{e} - x = 0$$
Subtracting $(1)$ from $(2)$, we get
$$SA'-SA=e(A'Z-AZ)$$ $$(CA'+CS)-(CA-CS)=e(AA')$$ $$2CS=e(AA')$$ $$CS=ae$$
Therefore, Coordinates of focus of ellipse $S$ is $(ae,0)$.
Now, from the definition of ellipse, $SP=e(PM)$
On squaring both sides, we get $$S{P^2} = {e^2}(P{M^2})$$
We can calculate the value of $SP$ using distance formulae between two points and the value of $PM$ using the perpendicular distance from a point to line i.e., from $P(x,y)$ to directrix $x = \frac{a}{e}$.
$$\begin{equation} \begin{aligned} {(x - ae)^2} + {(y - 0)^2} = {e^2}{(\frac{a}{e} - x)^2} \\ {x^2} + {a^2}{e^2} - 2aex + {y^2} = {a^2} - 2aex + {e^2}{x^2} \\ {x^2}(1 - {e^2}) + {y^2} = {a^2}(1 - {e^2}) \\ \frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{a^2}(1 - {e^2})}} = 1 \\ \frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1 \\\end{aligned} \end{equation} $$
where ${b^2} = {a^2}(1 - {e^2})$ (In this case $a>b$).
