Equation of a circle

The circle passing through the point $(-1,0)$ and touching the $y$-axis at $(0,2)$ also passes through the point

The straight line $2x-3y=1$ divides the circular region $x^2+y^2\le6$ into two parts. If
$$S=\left\{\left(2,\frac34\right),\left(\frac52,\frac34\right),\left(\frac14,-\frac14\right),\left(\frac18,\frac14\right)\right\},$$
then the number of point(s) in $S$ lying inside the smaller part is

Let A and B be two distinct points on the parabola $y^2=4x$. If the axis of the parabola touches a circle of radius r having AB as its diameter, then the slope of the line joining A and B can be

Equation of the circle with AB as its diameter is

Two parallel chords of a circle of radius 2 are at a distance $\sqrt3+1$ apart. If the chords subtend at the center, angles of $\dfrac{\pi}{k}$ and $\dfrac{2\pi}{k}$, where $k>0$, then the value of $[k]$ is

[Note: $[k]$ denotes the largest integer less than or equal to $k$]

Tangents drawn from the point $P(1,8)$ to the circle $$x^2+y^2-6x-4y-11=0$$ touch the circle at the points $A$ and $B$. The equation of the circumcircle of the triangle $PAB$ is

The line passing through the extremity $A$ of the major axis and extremity $B$ of the minor axis of the ellipse
$$x^2+9y^2=9$$
meets its auxiliary circle at the point $M$. Then the area of the triangle with vertices at $A$, $M$ and the origin $O$ is

The centres of two circles $C_1$ and $C_2$ each of unit radius are at a distance of 6 units from each other. Let $P$ be the mid point of the line segment joining the centres of $C_1$ and $C_2$ and $C$ be a circle touching circles $C_1$ and $C_2$ externally. If a common tangent to $C_1$ and $C$ passing through $P$ is also a common tangent to $C_2$ and $C$, then the radius of the circle $C$ is

Consider the two curves
$$C_1:y^2=4x,\qquad C_2:x^2+y^2-6x+1=0.$$
Then,

Let $a$ and $b$ be non-zero real numbers. Then, the equation
$$(ax^2+by^2+c)(x^2-5xy+6y^2)=0$$
represents

A straight line through the vertex $P$ of a triangle $PQR$ intersects the side $QR$ at the point $S$ and the circumcircle of the triangle $PQR$ at the point $T$. If $S$ is not the centre of the circumcircle, then

The equation of circle $C$ is

The number of elements in the set $A\cap B\cap C$ is

Consider
$$L_1:2x+3y+p-3=0,\qquad L_2:2x+3y+p+3=0,$$
where $p$ is a real number, and $C:x^2+y^2+6x-10y+30=0$.

STATEMENT-1: If line $L_1$ is a chord of circle $C$, then line $L_2$ is not always a diameter of circle $C$.

and

STATEMENT-2: If line $L_1$ is a diameter of circle $C$, then line $L_2$ is not a chord of circle $C$.

The differential equation $\dfrac{dy}{dx}=\dfrac{\sqrt{1-y^2}}{y}$ determines a family of circles with