The eccentricity of an ellipse, with its centre at the origin, is $\frac{1}{2}$. If one of the directrices is $x = 4$, then the equation of the ellipse is
The eccentricity of an ellipse, with its centre at the origin, is $\frac{1}{2}$. If one of the directrices is $x = 4$, then the equation of the ellipse is
- [AIEEE 2004]
- A
$4{x^2} + 3{y^2} = 1$
- B
$3{x^2} + 4{y^2} = 12$
- C
$4{x^2} + 3{y^2} = 12$
- D
$3{x^2} + 4{y^2} = 1$
Similar Questions
The distance between the foci of the ellipse $3{x^2} + 4{y^2} = 48$ is
The distance between the foci of the ellipse $3{x^2} + 4{y^2} = 48$ is
The centre of an ellipse is $C$ and $PN$ is any ordinate and $A$, $A’$ are the end points of major axis, then the value of $\frac{{P{N^2}}}{{AN\;.\;A'N}}$ is
The centre of an ellipse is $C$ and $PN$ is any ordinate and $A$, $A’$ are the end points of major axis, then the value of $\frac{{P{N^2}}}{{AN\;.\;A'N}}$ is
An ellipse has eccentricity $\frac{1}{2}$ and one focus at the point $P\left( {\frac{1}{2},\;1} \right)$. Its one directrix is the common tangent nearer to the point $P$, to the circle ${x^2} + {y^2} = 1$ and the hyperbola ${x^2} - {y^2} = 1$. The equation of the ellipse in the standard form, is
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- [IIT 1996]
Consider two straight lines, each of which is tangent to both the circle $x ^2+ y ^2=\frac{1}{2}$ and the parabola $y^2=4 x$. Let these lines intersect at the point $Q$. Consider the ellipse whose center is at the origin $O (0,0)$ and whose semi-major axis is $OQ$. If the length of the minor axis of this ellipse is $\sqrt{2}$, then which of the following statement($s$) is (are) $TRUE$?
$(A)$ For the ellipse, the eccentricity is $\frac{1}{\sqrt{2}}$ and the length of the latus rectum is $1$
$(B)$ For the ellipse, the eccentricity is $\frac{1}{2}$ and the length of the latus rectum is $\frac{1}{2}$
$(C)$ The area of the region bounded by the ellipse between the lines $x=\frac{1}{\sqrt{2}}$ and $x=1$ is $\frac{1}{4 \sqrt{2}}(\pi-2)$
$(D)$ The area of the region bounded by the ellipse between the lines $x=\frac{1}{\sqrt{2}}$ and $x=1$ is $\frac{1}{16}(\pi-2)$
Consider two straight lines, each of which is tangent to both the circle $x ^2+ y ^2=\frac{1}{2}$ and the parabola $y^2=4 x$. Let these lines intersect at the point $Q$. Consider the ellipse whose center is at the origin $O (0,0)$ and whose semi-major axis is $OQ$. If the length of the minor axis of this ellipse is $\sqrt{2}$, then which of the following statement($s$) is (are) $TRUE$?
$(A)$ For the ellipse, the eccentricity is $\frac{1}{\sqrt{2}}$ and the length of the latus rectum is $1$
$(B)$ For the ellipse, the eccentricity is $\frac{1}{2}$ and the length of the latus rectum is $\frac{1}{2}$
$(C)$ The area of the region bounded by the ellipse between the lines $x=\frac{1}{\sqrt{2}}$ and $x=1$ is $\frac{1}{4 \sqrt{2}}(\pi-2)$
$(D)$ The area of the region bounded by the ellipse between the lines $x=\frac{1}{\sqrt{2}}$ and $x=1$ is $\frac{1}{16}(\pi-2)$
- [IIT 2018]
Find the equation for the ellipse that satisfies the given conditions: Vertices $(\pm 6,\,0),$ foci $(\pm 4,\,0)$
Find the equation for the ellipse that satisfies the given conditions: Vertices $(\pm 6,\,0),$ foci $(\pm 4,\,0)$