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# Find the equation of the ellipse if the centre at the origin, the major axis is along $x$-axies,$e=\large\frac{2}{3}$ and passes through the point $(2 , \large\frac{-5}{3})$

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A)
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}$$=1 • http://clay6.com/mpaimg/1_Toolbox.png • Foci(\pm ae,o), center (0,0),vertices (\pm a,0) • End points of Latus Rectum (ae,\pm \large\frac{b^2}{a}) and (-ae,\pm \large\frac{b^2}{a}) • Directrices x=\pm \large\frac{a}{e}. • The major axis is y=0 (x- axis) and the minor axes is x=0 (y- axis) • \large\frac{x^2}{b^2}+\frac{y^2}{a^2}$$=1$
• http://clay6.com/mpaimg/1_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)
Step 1:
The centre is at $(0,0)$,major axis along $x$-axis,$e=\large\frac{2}{3}$
The equation is of the form $\large\frac{x^2}{a^2}+\frac{y^2}{b^2}$$=1-----(1) Where b^2=a^2(1-e^2) \Rightarrow a^2\big(1-\large\frac{4}{9}\big)=\large\frac{5a^2}{9} Substitute this in eq(1) we get \large\frac{x^2}{a^2}+\frac{9y^2}{5a^2}$$=1$
Step 2:
The point $(2,-\large\frac{5}{3}\big)$ lies on the ellipse .
Substituting the coordinates we have
$\large\frac{4}{a^2}+\large\frac{9\times 25/7}{5a^2}$$=1 \large\frac{1}{a^2}$$[4+5]=1$
$a^2=9$
$a= 3$
$b=\large\frac{\sqrt 5\times 3}{3}$$\Rightarrow \sqrt 5 The equation of the ellipse is \large\frac{x^2}{9}+\frac{y^2}{5}$$=1$
This is the required equation.