The locus of the centers of the circles such that the point $(2, 3)$ is the midpoint of the chord $5x + 2y = 16$ is:

Two concentric circles are such that the smaller divides the larger into two regions of equal area. If the radius of the smaller circle is $2$,then the length of the tangent from any point $P$ on the larger circle to the smaller circle is:

$A$ common tangent to $9x^2 + 16y^2 = 144$,$y^2 - x + 4 = 0$,and $x^2 + y^2 - 12x + 32 = 0$ is:

The locus of the midpoints of the chords of the circle $x^2 + y^2 - ax - by = 0$ which subtend a right angle at $\left( \frac{a}{2}, \frac{b}{2} \right)$ is:

If $\theta$ is the acute angle between the curves $x^2+y^2=2020 \sqrt{2}$ and $x^2-y^2=2020$, then $\frac{\sin \theta+\cos \theta}{\tan \theta}$ is equal to

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 $4\sqrt{2}$