$A$ rectangular hyperbola passing through $(3,2)$ has its asymptotes parallel to the coordinate axes. If $(1,1)$ is the point of intersection of the two perpendicular tangents of that hyperbola,then its equation is

Let $H: \frac{-x^2}{a^2}+\frac{y^2}{b^2}=1$ be the hyperbola,whose eccentricity is $\sqrt{3}$ and the length of the latus rectum is $4\sqrt{3}$. Suppose the point $(\alpha, 6)$,where $\alpha > 0$,lies on $H$. If $\beta$ is the product of the focal distances of the point $(\alpha, 6)$,then $\alpha^2+\beta$ is equal to:

The locus of the midpoints of the chord of the circle $x^{2}+y^{2}=25$ which is tangent to the hyperbola $\frac{x^{2}}{9}-\frac{y^{2}}{16}=1$ is

The eccentricity of a rectangular hyperbola is

If $5x + 9 = 0$ is the directrix of the hyperbola $16x^2 - 9y^2 = 144,$ then its corresponding focus is

If the circle $x^2+y^2=a^2$ intersects the hyperbola $xy=c^2$ in four points $(x_i, y_i)$,for $i=1, 2, 3, 4$,then $y_1+y_2+y_3+y_4$ equals