The angle of intersection between the curves $y^2+x^2=a^2 \sqrt{2}$ and $x^2-y^2=a^2$ is

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

If the length of the tangent from $(h, k)$ to the circle $x^2+y^2=16$ is twice the length of the tangent from the same point to the circle $x^2+y^2+2x+2y=0$,then:

If $(2, a)$ does not lie outside the circles $x^2+y^2=13$ and $x^2+y^2+x-2y=14$,then $a$ lies in

The equation of a common tangent to the circle $x^2+y^2=4$ and the ellipse $2x^2+25y^2=50$ is

The circle $C_1: x^2+y^2=3$,with centre at $O$,intersects the parabola $x^2=2y$ at the point $P$ in the first quadrant. Let the tangent to the circle $C_1$ at $P$ touch two other circles $C_2$ and $C_3$ at $R_2$ and $R_3$,respectively. Suppose $C_2$ and $C_3$ have equal radii $2\sqrt{3}$ and centres $Q_2$ and $Q_3$,respectively. If $Q_2$ and $Q_3$ lie on the $y$-axis,then:
$(A)$ $Q_2Q_3=12$
$(B)$ $R_2R_3=4\sqrt{6}$
$(C)$ Area of the triangle $OR_2R_3$ is $6\sqrt{2}$
$(D)$ Area of the triangle $PQ_2Q_3$ is $4\sqrt{2}$