If the lines $kx + 2y - 4 = 0$ and $5x - 2y - 4 = 0$ are conjugate with respect to the circle $x^2 + y^2 - 2x - 2y + 1 = 0$,then $k$ is equal to

Prove that the curves $y^{2}=4x$ and $x^{2}+y^{2}-6x+1=0$ touch each other at the point $(1,2)$.

Let $AB$ be a chord of the circle $x^2 + y^2 = r^2$ subtending a right angle at the centre. Then the locus of the centroid of the $\Delta PAB$ as $P$ moves on the circle is

For the circle $C$ with the equation $x^2+y^2-16x-12y+64=0$,match the List-$I$ with the List-$II$ given below.

List-$I$	List-$II$
$(i)$ The equation of the polar of $(-5, 1)$ with respect to $C$	$(A)$ $y = 0$
$(ii)$ The equation of the tangent at $(8, 0)$ to $C$	$(B)$ $y = 6$
$(iii)$ The equation of the normal at $(2, 6)$ to $C$	$(C)$ $x + y = 7$
$(iv)$ The equation of the diameter of $C$ through $(8, 12)$	$(D)$ $13x + 5y = 98$
	$(E)$ $x = 8$

The correct match is:

For the given circle $2x^2 + 2y^2 = 5$ and the parabola $y^2 = 4\sqrt{5}x$:
Statement-$I$: The equation of the common tangent to these curves is $y = x + \sqrt{5}$.
Statement-$II$: If the line $y = mx + \frac{\sqrt{5}}{m} (m \neq 0)$ is a common tangent,then $m$ satisfies $m^4 - 3m^2 + 2 = 0$.

An equilateral triangle $OAB$ is inscribed in the parabola $y^{2}=4x$ with the vertex $O$ at the origin. Then the minimum distance of the circle having $AB$ as a diameter from the origin is: