The line $x=2y$ intersects the ellipse $\frac{x^{2}}{4}+y^{2}=1$ at the points $P$ and $Q$. The equation of the circle with $PQ$ as diameter is

Consider the equations of the circles:
$S_1 : x^2 + y^2 + 24x - 10y + a = 0$
$S_2 : x^2 + y^2 = 36$
Which of the following statements is not correct?

The sum of diameters of the circles that touch $(i)$ the parabola $75x^2 = 64(5y - 3)$ at the point $\left(\frac{8}{5}, \frac{6}{5}\right)$ and $(ii)$ the $y$-axis,is equal to $......$

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 tangent at $A(-1, 2)$ on the circle $x^2+y^2-4x-8y+7=0$ touches the circle $x^2+y^2+4x+6y=0$ at $B$. Then,a point of trisection of $AB$ 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 $4\sqrt{2}$