Let a variable circle touches a line, which is tangent to the parabola $y^2=4x$ at  (4, 4) and variable circle passes through focus of the parabola $y^2=4x$. If locus of its centre is conic ‘C’, then then which of the following is/are true?

Equation of tangent to $y^2=4x$ at (4,4) is x – 2y + 4 = 0

Suppose that the center of the circle is $P\left(x_1,y_1\right)$

The distance from $P\left(x_1,y_1\right)$ to the above line is equal to the distance from $P\left(x_1,y_1\right)$ to the focus of the parabola

hence, 

$\frac{x_1-2y_1+4}{\sqrt{1+4}}=\sqrt{{\left(x_1-1\right)}^2+{\left(y_1-0\right)}^2}$

The above equation is the parabola having focus at $\left(1,0\right)$ and directrix $x-2y+4=0$

Axis of the conic is a line perpendicular to the above line and passing through the focus

it is 2x+y=2

Hence the length of the latusrectum is =$2\times \frac{5}{\sqrt{5}}=2\sqrt{5}$