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 $\small \frac{3}{5}$ 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, AA’ = 2a and BB’ = 2b

The foci are S (0, be) and S’(0, be)

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

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	$\mathrm{x}=\pm \mathrm{a}$	$y=\pm b$