If a variable line,$3x + 4y - \lambda = 0$,is such that the two circles $x^2 + y^2 - 2x - 2y + 1 = 0$ and $x^2 + y^2 - 18x - 2y + 78 = 0$ are on its opposite sides,then the set of all values of $\lambda$ is the interval

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 $4\sqrt{2}$

If the portion of the line $lx + my = 1$ falling inside the circle ${x^2} + {y^2} = {a^2}$ subtends an angle of $45^\circ$ at the origin,then

Two tangents $PQ$ and $PR$ are drawn to the circle $x^2 + y^2 - 2x - 4y - 20 = 0$ from the point $P(16, 7)$. If the centre of the circle is $C$,then the area of quadrilateral $PQCR$ is ............ $sq. \text{ units}$.

Let the parabola $y=x^2+px-3$ meet the coordinate axes at the points $P, Q$ and $R$. If the circle $C$ with center at $(-1,-1)$ passes through the points $P, Q$ and $R$,then the area of $\triangle PQR$ is:

Let $AB$ be a chord of the circle $x^2 + y^2 = r^2$ subtending a right angle at the centre. Then the locus of the centroid of the $\Delta PAB$ as $P$ moves on the circle is