If the two circles,$x^2 + y^2 + 2g_1x + 2f_1y = 0$ and $x^2 + y^2 + 2g_2x + 2f_2y = 0$ touch each other,then:

If the equation of the circumcircle of the triangle formed by the lines $L_1 \equiv x+y=0$,$L_2 \equiv 2x+y-1=0$,and $L_3 \equiv x-3y+2=0$ is $\lambda_1 L_1 L_2 + \lambda_2 L_2 L_3 + \lambda_3 L_3 L_1 = 0$,then find the value of $\frac{7 \lambda_1}{\lambda_2} + \frac{\lambda_3}{\lambda_1}$.

If the tangent at the point $\left(4 \cos 2 \theta, \frac{16}{\sqrt{11}} \sin 2 \theta\right)$ on the ellipse $16 x^2+11 y^2=256$ touches the circle $x^2+y^2-2 x=15$,then $\theta=$

Given: $A$ circle $2x^2 + 2y^2 = 5$ and a parabola $y^2 = 4\sqrt{5}x$.
Statement-$1$: An equation of a common tangent to these curves is $y = x + \sqrt{5}$.
Statement-$2$: If the line $y = mx + \frac{\sqrt{5}}{m} (m \neq 0)$ is their common tangent,then $m$ satisfies $m^4 - 3m^2 + 2 = 0$.

If the point $(1,4)$ lies inside the circle $x^2+y^2-6x-10y+p=0$ and the circle does not touch or intersect the coordinate axes,then

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 $(1, 3)$	$IV$. $x=9$