The line $lx + my + n = 0$ will be a tangent to the circle $x^2 + y^2 = a^2$ if

The area of the triangle formed by the positive $X$-axis, the tangent and the normal to the curve $x^2+y^2=16a^2$ at the point $(2\sqrt{2}a, 2\sqrt{2}a)$ is

Consider the circle $S: x^2 + y^2 = 1$ and point $P(0, -1)$ on it. $A$ ray of light passes through the point $(-3, -1)$ and reflects from the tangent to $S$ at $P$. After reflection,it becomes tangent to the circle $S$. Find the equation of the reflected ray.

$O(0,0)$ and $A(1,0)$ are the centers of two unit circles $C_1$ and $C_2$ respectively. $C_3$ is also a unit circle having its center above the $X$-axis and passing through $O$ and $A$. The equation of the common tangent to $C_1$ and $C_3$ which does not intersect the circle $C_2$ is

If the tangent and the normal at the point $(\sqrt{3}, 1)$ to the circle $x^2+y^2=4$,and the $X$-axis form a triangle,then the area (in sq. units) of this triangle is

If $m_1, m_2$ are the slopes of the tangents drawn from a point $(1, -3)$ to the circle $x^2+y^2-6x+4y+12=0$,then $9(m_1^2+m_2^2) = $