Definitions and Terms In Ellipse

Find the equation of the directrix of the ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1$

Find the equation of the directrix of the ellipse $\frac{x^2}{100}+\frac{y^2}{36}=1$

Find the equation of the diameter of an ellipse $2x^2+3y^2=6$ conjugate to the diameter $3y+2x=0$

Find the equation of the diameter of an ellipse $3x^2+4y^2=5$ conjugate to the diameter $y+3x=0$

Let $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(b<a),$ be an ellipse with major axis $AB$ and minor axis $CD.$ Let $F_1$ and $F_2$ be its two foci, with $A,F_1, F_2, B$ in that order on the segment $AB.$ Suppose $\angle F_{1}CB=90°.$ The eccentricity of the ellipse is

If a tangent to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, whose centre is $C$, meets the major and the minor axes at $M$ and $N$ respectively then $\frac{a^2}{C{M}^2}+\frac{b^2}{C{N}^2}$ is equal to

Find the equation of the ellipse in the standard form whose distance between foci is $2$ and the length of latus rectum is $\frac{15}{2}$.

S and T are the foci of the ellipse $\left(\frac{x^2}{a^2}\right)+\left(\frac{y^2}{b^2}\right)=1$ and B is an end of the minor axis. If STB is an equilateral triangle, then eccentricity of the ellipse is

Let $x^2=4ky,k>0$ be a parabola with vertex $A$. Let $BC$ be its latusrectum. An ellipse with centre on $BC$ touches the parabola at $A$, and cuts $BC$ at points $D$ and $E$ such that $BD=DE=EC$ ($B, D, E, C$ in that order). The eccentricity of the ellipse is

Let $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,a>b$ be an ellipse with foci $F_1$ and $F_2$ Let $AO$ be its semi-minor axis, where $O$ is the centre of the ellipse. The lines $A{F}_1$ and $A{F}_2$, when extended, cut the ellipse again at points $B$ and $C$ respectively. Suppose that the $\mathrm{\Delta }ABC$ is equilateral. Then, the eccentricity of the ellipse is

If the latus rectum of an ellipse is equal to half of minor axis, then find the value of $k$, if its eccentricity is $\frac{\sqrt{k}}{2}$

The eccentricity of an ellipse if it has $OB$ as semi-minor axis, $F$ and $F^{\text{'}}$ as it's foci and the angle $FB{F}^{\prime }$ is a right angle is:

The eccentricity of ellipse $14x^2-4xy+11y^2=60$ will be equal to 

If $a\left(x^2+y^2+2y+1\right)={\left(x-2y+3\right)}^2$ is an ellipse and $a\in \left(b,\infty \right),$ then the value of $b$ is _________.

A parabola is drawn with focus at one of the foci of the ellipse $\frac{x^2}{a}+\frac{y^2}{b}=1$ , where $a>b$, and directrix passing through the other focus and perpendicular to the major axis of the ellipse. If the latus rectum of the ellipse and that of the parabola are same, then the eccentricity of the ellipse is:

An ellipse passing through $\left(4\sqrt{2}, 2\sqrt{6}\right)$ has foci at $\left(-4, 0\right)$ and $\left(4, 0\right)$ . Then, its eccentricity is

If $\left(x_1, y_1\right)$ is a point on the ellipse $b^2 x^2+a^2 y^2=a^2 b^2$ then the area of $\mathrm{\Delta }SP{S}^{\prime }=$

In an ellipse the distance between its foci is $6$ and its minor axis is $8$. Then its eccentricity is

The length of the latus-rectum of the ellipse $5{\mathrm{x}}^2+9{\mathrm{y}}^2=\mathrm{45 }$ is

$AB$ is a diameter of $x^2+ 9y^2=25$ . The eccentric angle of $A$ is $\pi /6$. Then the eccentric angle of $B$ is -