If the perpendicular distance from the focus of a parabola $y^2=4 a x$ to its directrix is $\frac{3}{2}$, then the equation of the normal drawn at $(4 a,-4 a)$ is

Let $A_1$ be the area of the given ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

Let $A_2$ be the area of the region bounded by the curve which is the locus of mid-point of the line segment joining the focus of the ellipse and a point $P$ on the given ellipse, then $A_1: A_2=$

If the equation of the tangent of the hyperbola $5 x^2-9 y^2-20 x-18 y-34=0$ which makes an angle $45^{\circ}$ with the positive $X$-axis in positive direction is $x+b y+c=0$, then $b^2+c^2=$

If the distance between the foci of a hyperbola $H$ is 26 and distance between its directrices is $\frac{50}{13}$, then the eccentricity of the conjugate hyperbola of the hyperbola $H$ is