If the latus rectum of a hyperbola subtends an angle of $120^{\circ}$ at its centre,then its eccentricity is

Let the foci of a hyperbola $H$ coincide with the foci of the ellipse $E: \frac{(x-1)^2}{100}+\frac{(y-1)^2}{75}=1$ and the eccentricity of the hyperbola $H$ be the reciprocal of the eccentricity of the ellipse $E$. If the length of the transverse axis of $H$ is $\alpha$ and the length of its conjugate axis is $\beta$,then $3 \alpha^2+2 \beta^2$ is equal to :

Let $P$ be a point on the hyperbola $H: \frac{x^2}{9}-\frac{y^2}{4}=1$,in the first quadrant such that the area of the triangle formed by $P$ and the two foci of $H$ is $2 \sqrt{13}$. Then,the square of the distance of $P$ from the origin is

If the product of the slopes of the tangents drawn from an external point $P(h, k)$ to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is a constant $k^2$,then the locus of $P$ is

Let a line $L_{1}$ be tangent to the hyperbola $\frac{x^{2}}{16}-\frac{y^{2}}{4}=1$ and let $L_{2}$ be the line passing through the origin and perpendicular to $L_{1}$. If the locus of the point of intersection of $L_{1}$ and $L_{2}$ is $(x^{2}+y^{2})^{2} = \alpha x^{2}+\beta y^{2}$,then $\alpha+\beta$ is equal to

If $ax^{2}+2hxy+by^{2}+2gx+2fy+c=0$ represents a joint equation of the directrices of the hyperbola $16x^{2}-9y^{2}=144$,then $g+f-c=$