Interaction Between Two Circles

The radius of the circle, having centre at (2, 1), whose one of the chord is a diameter of the circle ${\text{x}}^2+{\text{y}}^2-2\text{x}-6\text{y}+6=0$

Let $C_1$ be the circle of radius $1$ with center at the origin. Let $C_2$ be the circle of radius $r$ with center at the point $A=(4,1)$, where $1<r<3$. Two distinct common tangents $PQ$ and $ST$ of $C_1$ and $C_2$ are drawn. The tangent $PQ$ touches $C_1$ at $P$ and $C_2$ at $Q$. The tangent $ST$ touches $C_1$ at $S$ and $C_2$ at $T$. Midpoints of the line segments $PQ$ and $ST$ are joined to form a line which meets the $x$-axis at a point $B$. If $AB=\sqrt{5}$, then the value of $r^2$ is

Consider two circles $C_1:{\left(x-1\right)}^2+{\left(y-4\right)}^2-16=0$ and $C_2:{\left(x-13\right)}^2+{\left(y-9\right)}^2-81=0$. If a circle of radius $r$ touches $x-$axis and $C_1$ and $C_2$ externally, then $r$ is equal to

In the figure given below with centre $O$, and another circle with centre at the mid-point $B$ of the radius $OA$ and passing through $O$. A small circle is drawn as shown, tangent to both the larger circles such that its centre $N$ is directly above $B$(that is $BN\perp AB$). If the perimeter of the triangle $NBA$ is $8$ cm, find the value in $cm$, of the radius of the small circle.

Three circles $C_3, C_2, C_1$ of radius $3, 6, 9$ respectively touch as shown in figure. A chord $AB$ to circle $C_1$ touch circle $C_2 \& C_3$.

The length of chord $AB$ is $\alpha \sqrt{14}$ and length of common tangent $PQ$ is $\beta \sqrt{2}$, then number of divisors of $\left(3^{\beta -1}-2^{\alpha -1}\right)$ is

If the circles $a{x}^2+a{y}^2+2bx+2cy=0$ and $A{x}^2+A{y}^2+2Bx+2Cy=0$ touch each other then

Let $G$ be a circle of radius $R>0$. Let $G_1,G_2,\dots ,G_n$ be $n$ circles of equal radius $r>0$. Suppose each of the $n$ circles $G_1,G_2,\dots ,G_n$ touches the circle $G$ externally. Also, for $i=1,2,\dots ,n-1$, the circle $G_i$ touches $G_{i+1}$ externally, and $G_n$ touches $G_1$ externally. Then, which of the following statements is/are TRUE?

Let $ABC$ be the triangle with $AB=1,AC=3$ and $\angle BAC=\frac{\pi }{2}$. If a circle of radius $r>0$ touches the sides $AB,AC$ and also touches internally the circumcircle of the triangle $ABC$, then the value of $r$ is ______.

(If the numerical value has more than two decimal places, truncate/round-off the value to TWO decimal places)

Which of the following statements is true about circle $x^2+y^2+2x=0$ and $x^2+y^2-6x=0$

Circle with centres $O$ and $P$ have radii $2$ and $4$ units respectively, and are externally tangent. Points $A$ and $B$ are on the circle centred at $O,$ and points $C$ and $D$ are on the circle centred at $P,$ such that $\overline{AD}$ and $\overline{BC}$ are common external tangents to the circles. What is the area of hexagon $AOBCPD$?

The centres of the three circles $A, B,$ and $C$ are collinear with the centre of circle $B$ lying between the centres of circle $A$ and $C$. Circles $A$ and $C$ both touch externally to circle $B$ and the three circles share a common tangent line. Given that circle $A$ has radius $12$ and circle $B$ has radius $42$ then the radius of circle $C$ is equal to

The common tangents to the circles $x^2+y^2+6x=0$ and $x^2+y^2-2x=0$ forms a triangle with centroid $G$, circum-centre $S$ and in-centre $I$. Then

If the radical centre of the following circles: $x^2+y^2+4x-7=0, 2x^2+2y^2+3x+5y-9=0, x^2+y^2+y=0$ is $\left(a,b\right)$ then find the value of $a+b$.

The equation of the radical axis of the circles: $x^2+y^2+2x+4y+1=0$ and $x^2+y^2+4x+y=0$ is

The equation of the radical axis of the circles: $x^2+y^2-3x-4y+5=0,$ and $3\left(x^2+y^2\right)-7x+8y-11=0$ is

The equation of the circle which cuts orthogonally the circle $x^2+y^2-4x+2y-7=0$ and having the centre at $(2,3)$ is

The equation of the circle which passes through the points $(2,0),(0,2)$ and orthogonal to the circle $2x^2+2y^2+5x-6y+4=0$ is

The equation of the circle passing through the origin, having its centre on the line $x+y=4$ and intersecting the circle $x^2+y^2-4x+2y+4=0$ orthogonally is

The equation of the circle which passes through the origin and intersects the circles, $x^2+y^2-4x-6y-3=0,$ and $x^2+y^2-8y+12=0$ orthogonally is