Horizontal and Vertical Ellipse MCQ - Practice Questions & Answers

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Consider an ellipse, whose centre is at the origin and its major axis is along the $x$-axis. If its eccentricity is $\frac{3}{5}$ and the distance between its foci is 6 , then the area (in sq. units) of the quadrilateral inscribed in the ellipse, with the vertices as the end points of major and minor axes of ellipse, is

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The equation of the ellipse whose axes are the axes of coordinates and which passes through the point (–3, 1) and has eccentricity $\sqrt{\frac{2}{5}}$ is

$3x^{2}+5y^{2}-15=0\;$

$\; 5x^{2}+3y^{2}-32=0\;$

$\; 3x^{2}+5y^{2}-32=0\; \;$

$\; 5x^{2}+3y^{2}-54=0$

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Concepts Covered - 1

Horizontal and Vertical Ellipse

Horizontal and Vertical Ellipse

When the major axis is along Y -axis and the minor axis is along X -axis, i.e. b > a
Then, $\mathrm{AA}^{\prime}=2 \mathrm{a}$ and $\mathrm{BB}^{\prime}=2 \mathrm{~b}$
The foci are $\mathrm{S}(0$, be $)$ and $S^{\prime}(0$, be $)$

The equation of directrix MZ and M’Z’ are $y=\frac{b}{e} \quad$ and $y=-\frac{b}{e}$

Equation	$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 ; a>b$	$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 ; a<b$
Graph
Centre	(0 ,0)	(0, 0)
Vertices	$(\pm \mathrm{a}, 0)$	$(0, \pm \mathrm{b})$
Length of Major Axis	2a	2b
Length of Minor Axis	2b	2a
Foci	$(\pm \mathrm{ae}, 0)$	$(0, \pm \mathrm{be})$
Distance b/w foci	2ae	2be
Equation of Directrices	$\mathrm{x}=\pm \frac{\mathrm{a}}{\mathrm{e}}$	$y=\pm\frac{b}{e}$
Distance b/w Directrices	$\frac{2 \mathrm{a}}{\mathrm{e}}$	$\frac{2 \mathrm{b}}{\mathrm{e}}$
Eccentricity, e	$\mathrm{e}=\sqrt{1-\frac{\mathrm{b}^{2}}{\mathrm{a}^{2}}}$	$\mathrm{e}=\sqrt{1-\frac{\mathrm{a}^{2}}{\mathrm{b}^{2}}}$
Length of Latusrectum	$\frac{2 \mathrm{b}^{2}}{\mathrm{a}}$	$\frac{2 \mathrm{a}^{2}}{\mathrm{b}}$
Endpoint of Latusrectum	$\left(\pm \mathrm{ae}, \pm \frac{\mathrm{b}^{2}}{\mathrm{a}}\right)$	$\left(\pm \frac{\mathrm{a}^{2}}{\mathrm{b}}, \pm \mathrm{be}\right)$
Focal radii	SP+S'P = 2a	SP+S'P = 2b
Parametric coordinates	${\color{Blue} (a \cos \theta, b \sin \theta) \quad 0 \leq \theta \leq 2 \pi }$	${\color{Blue} (a \cos \theta, b \sin \theta) \quad 0 \leq \theta \leq 2 \pi }$
Tangent at vertices	$\mathrm{x}=\pm \mathrm{a}$	$y=\pm b$