Find the coordinates of the foci,the vertices,the length of the major axis,the minor axis,the eccentricity,and the length of the latus rectum of the ellipse $4x^{2} + 9y^{2} = 36$.
(A) The given equation is $4x^{2} + 9y^{2} = 36$.
Dividing both sides by $36$,we get:
$\frac{x^{2}}{9} + \frac{y^{2}}{4} = 1$
$\frac{x^{2}}{3^{2}} + \frac{y^{2}}{2^{2}} = 1$
Comparing this with the standard equation $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$,where $a > b$,we have $a = 3$ and $b = 2$.
The eccentricity $e$ is given by $e = \sqrt{1 - \frac{b^{2}}{a^{2}}} = \sqrt{1 - \frac{4}{9}} = \sqrt{\frac{5}{9}} = \frac{\sqrt{5}}{3}$.
The foci are $(\pm ae, 0) = (\pm 3 \times \frac{\sqrt{5}}{3}, 0) = (\pm \sqrt{5}, 0)$.
The vertices are $(\pm a, 0) = (\pm 3, 0)$.
The length of the major axis is $2a = 2 \times 3 = 6$.
The length of the minor axis is $2b = 2 \times 2 = 4$.
The length of the latus rectum is $\frac{2b^{2}}{a} = \frac{2 \times 4}{3} = \frac{8}{3}$.
Dividing both sides by $36$,we get:
$\frac{x^{2}}{9} + \frac{y^{2}}{4} = 1$
$\frac{x^{2}}{3^{2}} + \frac{y^{2}}{2^{2}} = 1$
Comparing this with the standard equation $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$,where $a > b$,we have $a = 3$ and $b = 2$.
The eccentricity $e$ is given by $e = \sqrt{1 - \frac{b^{2}}{a^{2}}} = \sqrt{1 - \frac{4}{9}} = \sqrt{\frac{5}{9}} = \frac{\sqrt{5}}{3}$.
The foci are $(\pm ae, 0) = (\pm 3 \times \frac{\sqrt{5}}{3}, 0) = (\pm \sqrt{5}, 0)$.
The vertices are $(\pm a, 0) = (\pm 3, 0)$.
The length of the major axis is $2a = 2 \times 3 = 6$.
The length of the minor axis is $2b = 2 \times 2 = 4$.
The length of the latus rectum is $\frac{2b^{2}}{a} = \frac{2 \times 4}{3} = \frac{8}{3}$.
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If $x+2y+k=0, k>0$ is a tangent to the ellipse $2x^2+y^2=2$,then the equation of the normal to the given ellipse at $\left(\frac{1}{\sqrt{2}}, \frac{k}{3}\right)$ is:
Define the collections $\{E_1, E_2, E_3, \ldots\}$ of ellipses and $\{R_1, R_2, R_3, \ldots\}$ of rectangles as follows:
$E_1: \frac{x^2}{9} + \frac{y^2}{4} = 1$
$R_1$: rectangle of largest area,with sides parallel to the axes,inscribed in $E_1$;
$E_n$: ellipse $\frac{x^2}{a_n^2} + \frac{y^2}{b_n^2} = 1$ of largest area inscribed in $R_{n-1}, n > 1$;
$R_n$: rectangle of largest area,with sides parallel to the axes,inscribed in $E_n, n > 1$.
Then which of the following options is/are correct?
$(1)$ The eccentricities of $E_{18}$ and $E_{19}$ are $NOT$ equal
$(2)$ The distance of a focus from the centre in $E_9$ is $\frac{\sqrt{5}}{32}$
$(3)$ The length of latus rectum of $E_9$ is $\frac{1}{6}$
$(4)$ $\sum_{n=1}^N (\text{area of } R_n) < 24$,for each positive integer $N$
$E_1: \frac{x^2}{9} + \frac{y^2}{4} = 1$
$R_1$: rectangle of largest area,with sides parallel to the axes,inscribed in $E_1$;
$E_n$: ellipse $\frac{x^2}{a_n^2} + \frac{y^2}{b_n^2} = 1$ of largest area inscribed in $R_{n-1}, n > 1$;
$R_n$: rectangle of largest area,with sides parallel to the axes,inscribed in $E_n, n > 1$.
Then which of the following options is/are correct?
$(1)$ The eccentricities of $E_{18}$ and $E_{19}$ are $NOT$ equal
$(2)$ The distance of a focus from the centre in $E_9$ is $\frac{\sqrt{5}}{32}$
$(3)$ The length of latus rectum of $E_9$ is $\frac{1}{6}$
$(4)$ $\sum_{n=1}^N (\text{area of } R_n) < 24$,for each positive integer $N$
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