If from any point on the circle $x^2+y^2+2gx+2fy+c=0$,tangents are drawn to the circle $x^2+y^2+2gx+2fy+c \sin^2 \alpha + (g^2+f^2) \cos^2 \alpha = 0$,where $0 < \alpha < \frac{\pi}{2}$,then the angle between those tangents is

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:
$(A) x^2+y^2-12x+24=0$
$(B) x^2+y^2+12x+24=0$
$(C) x^2+y^2+24x-12=0$
$(D) x^2+y^2-24x-12=0$

In a right triangle $ABC$,right-angled at $A$,a semicircle is described on the leg $AC$ as diameter. If $D$ is the point of intersection of the hypotenuse $BC$ and the semicircle,then the length $AC$ is equal to:

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 $\frac{7 \lambda_1}{\lambda_2} + \frac{\lambda_3}{\lambda_1}$.

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 $m^4 - 3m^2 + 2 = 0$.

$A$ variable line $ax + by + c = 0$,where $a, b, c$ are in $A.P.$,is normal to a circle $(x - \alpha)^2 + (y - \beta)^2 = \gamma$,which is orthogonal to the circle $x^2 + y^2 - 4x - 4y - 1 = 0$. The value of $\alpha + \beta + \gamma$ is equal to