An ellipse has $OB$ as semi-minor axis,$S$ and $S^{\prime}$ are foci and angle $\angle SBS^{\prime}$ is a right angle. Then the eccentricity of the ellipse is

The line $y=x+\lambda$ is tangent to the ellipse $2x^{2}+3y^{2}=1$. Then, $\lambda$ is

Let $E$ be an ellipse whose axes are parallel to the coordinate axes,having its center at $(3, -4)$,one focus at $(4, -4)$,and one vertex at $(5, -4)$. If $mx - y = 4$ with $m > 0$ is a tangent to the ellipse $E$,then the value of $5m^{2}$ is equal to .....

The lines $y=2x+\sqrt{76}$ and $2y+x=8$ touch the ellipse $\frac{x^2}{16}+\frac{y^2}{12}=1$. If the point of intersection of these two lines lies on a circle whose centre coincides with the centre of that ellipse,then the equation of that circle is

$S=(-1, 1)$ is the focus,$2x-3y+1=0$ is the directrix corresponding to $S$,and $\frac{1}{2}$ is the eccentricity of an ellipse. If $(a, b)$ is the centre of the ellipse,then $3a+2b=$

Let $E_1 = \frac{x^2}{9} + \frac{y^2}{4} = 1$ and $E_2 = \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ be two ellipses and $R$ be a rectangle with sides parallel to the coordinate axes. Let $E_1$ be the inscribed ellipse in $R$ and $E_2$ be the circumscribed ellipse on $R$. If $E_2$ passes through $(0, 4)$,then: