The length of the latus rectum of an ellipse is $\frac{1}{3}$ of the major axis. Its eccentricity is
$\frac{2}{3}$
$\sqrt {\frac{2}{3}} $
$\frac{{5 \times 4 \times 3}}{{{7^3}}}$
${\left( {\frac{3}{4}} \right)^4}$
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
Let $T_1$ and $T_2$ be two distinct common tangents to the ellipse $E: \frac{x^2}{6}+\frac{y^2}{3}=1$ and the parabola $P: y^2=12 x$. Suppose that the tangent $T_1$ touches $P$ and $E$ at the point $A_1$ and $A_2$, respectively and the tangent $T_2$ touches $P$ and $E$ at the points $A_4$ and $A_3$, respectively. Then which of the following statements is(are) true?
($A$) The area of the quadrilateral $A_1 A _2 A _3 A _4$ is $35$ square units
($B$) The area of the quadrilateral $A_1 A_2 A_3 A_4$ is $36$ square units
($C$) The tangents $T_1$ and $T_2$ meet the $x$-axis at the point $(-3,0)$
($D$) The tangents $T_1$ and $T_2$ meet the $x$-axis at the point $(-6,0)$
The locus of the point of intersection of perpendicular tangents to the ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$, is
An ellipse passes through the point $(-3, 1)$ and its eccentricity is $\sqrt {\frac{2}{5}} $. The equation of the ellipse is
On the ellipse $4{x^2} + 9{y^2} = 1$, the points at which the tangents are parallel to the line $8x = 9y$ are