If the focus divides a focal chord of the parabola $y^2 = 16x$ into $2$ parts having lengths $a$ and $c$,such that $a, b, c$ are in $H.P.$,then the value of $b$ is equal to:

Variable straight lines $y=mx+c$ make intercepts on the curve $y^2-4ax=0$ which subtend a right angle at the origin. Then the point of concurrence of these lines $y=mx+c$ is

The slope of the line touching both the parabolas $y^2 = 4x$ and $x^2 = -32y$ is

The equation of the normal to the parabola $y^2=4x$ which is perpendicular to the line $x+3y+1=0$ is

The point on the curve $y^2=2(x-3)$ at which the normal is parallel to the line $y-2x+1=0$ is

Consider the conic $C: 25(x - 1)^2 + 25(y + 1)^2 = (3x - 4y)^2$. If the curve $E$ is the locus of the point of intersection of perpendicular tangents to the conic $C$,then the minimum distance between the curve $E$ and the point $(2, -1)$ is: