# Find the equation of the ellipse if the foci are $(2 , -1 ) , (0 , -1 )$ and $e=\large\frac{1}{2}$

Toolbox:
• Standard forms of equation of the ellipse with major axes 2a, minor axis 2b $(a >b)$ eccentricity e and $b^2=a^2(1-e^2)$ or $e^2=1-\large\frac{b^2}{a^2}$
• $\large\frac{x^2}{a^2}+\frac{y^2}{b^2}$$=1 • http://clay6.com/mpaimg/3_Toolbox.png • Foci(\pm ae,o), center (0,0),vertices (\pm a,0) • End points of Latus Rectum (ae,\pm \large\frac{b^2}{a}) and (-ae,\pm \large\frac{b^2}{a}) • Directrices x=\pm \large\frac{a}{e}. • The major axis is y=0 (x- axis) and the minor axes is x=0 (y- axis) • \large\frac{x^2}{b^2}+\frac{y^2}{a^2}$$=1$
• http://clay6.com/mpaimg/3_Toolbox1.png
• Foci$(0;\pm ae),$ center $(0,0)$,vertices $(0,\pm a)$
• End points of Latus Rectum $(\pm \large\frac{b^2}{a}$$,ae) and (\pm \large\frac{b^2}{a}$$,-ae)$
• Directrices $y=\pm \large\frac{a}{e}$.
• The major axis is $x=0$ (y- axis) and the minor axes is $y=0$ (x- axis)
• General form of standard ellipses with centre $C$ with major axis $2a$,minor axis $2b$ and axes parallel to the coordinate axes.
• $\large\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}$$=1 • In this case major axis y=k and minor axis x=h. • \large\frac{(x-h)^2}{b^2}+\frac{(y-k)^2}{a^2}$$=1$
• In this case major axis $x=h$ and minor axis $y=k$.
Step 1:
$F(2,-1)$ and $F'(0,-1)$ are the foci.If $C$ is the centre,$CF=CF'=ac$.
Therefore $e$ is the midpoint of $FF'$.
$CF=1=ae$
$e$ is given to be $\large\frac{1}{2}$
$1=\large\frac{a}{2}$
$\Rightarrow a=2$
$b=a\sqrt{1-e^2}$
$\;\;=2\sqrt{1-\large\frac{1}{4}}$
$\;\;=\sqrt 3$
Step 2:
The major axis is along CF (i.e)$y=-1$[parallel to the x-axis].
The equation of the form
$\large\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}$$=1 Where (a,k) is the centre. \large\frac{(x-1)^2}{4}+\frac{(y+1)^2}{3}$$=1$
This is the required equation.
edited Jun 13, 2013