Tangents are drawn to a unit circle with centre at the origin from each point on the line $2x + y = 4$. Then the equation to the locus of the middle point of the chord of contact is

Given the circle $C$ with the equation $x^2+y^2-2x+10y-38=0$. Match the List-$I$ with the List-$II$ given below concerning $C$.

List-$I$	List-$II$
$A$. The equation of the polar of $(4, 3)$ with respect to $C$	$I$. $y+5=0$
$B$. The equation of the tangent at $(9, -5)$ on $C$	$II$. $x=1$
$C$. The equation of the normal at $(-7, -5)$ on $C$	$III$. $3x+8y=27$
$D$. The equation of the diameter passing through $(1, -5)$ and $(1, 3)$	$IV$. $x=9$

The locus of the centre of the circle which touches the circle ${x^2} + {(y - 1)^2} = 1$ externally and also touches the $x$-axis is

If the length of the tangent from any point on the circle $(x-3)^2+(y+2)^2=5r^2$ to the circle $(x-3)^2+(y+2)^2=r^2$ is $16$ units, then the area between the two circles in sq. units is (in $\pi$)

The circle $x^2+y^2-8x=0$ and hyperbola $\frac{x^2}{9}-\frac{y^2}{4}=1$ intersect at the points $A$ and $B$.
$1.$ Equation of a common tangent with positive slope to the circle as well as to the hyperbola is:
$(A) 2x-\sqrt{5}y-20=0$
$(B) 2x-\sqrt{5}y+4=0$
$(C) 3x-4y+8=0$
$(D) 4x-3y+4=0$
$2.$ Equation of the circle with $AB$ as its diameter is:
$(A) x^2+y^2-12x+24=0$
$(B) x^2+y^2+12x+24=0$
$(C) x^2+y^2+24x-12=0$
$(D) x^2+y^2-24x-12=0$

If a line along a chord of the circle $4x^{2}+4y^{2}+120x+675=0$ passes through the point $(-30, 0)$ and is tangent to the parabola $y^{2}=30x$,then the length of this chord is: