Circles

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 $T$. Mid points of the line segments $PQ$ and $ST$ are joined to form a line which meets the $x$-axis at a point $B$. If $AB = \sqrt5$, then the value of $r^2$ is _____.

Let $ABC$ be the triangle with $AB = 1$, $AC = 3$ and $\angle BAC = \dfrac\pi2$. 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 _____.

Let $G$ be a circle of radius $R > 0$. Let $G_1, G_2, \ldots, G_n$ be $n$ circles of equal radius $r > 0$. Suppose each of the $n$ circles $G_1, G_2, \ldots, G_n$ touches the circle $G$ externally. Also, for $i = 1, 2, \ldots, n-1$, 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 $x^2+y^2+Ax+By+C=0$ be a circle passing through $(0,6)$ and touching the parabola $y=x^2$ at $(2,4)$. Then $A+C$ is equal to ______.

Let $S=\{z\in\mathbb C:|z-3|\le1$ and $z(4+3i)+\bar z(4-3i)\le24\}$. If $\alpha+i\beta$ is the point in $S$ which is closest to $4i$, then $25(\alpha+\beta)$ is equal to ______.

Let a circle $C:(x-h)^2+(y-k)^2=r^2$, $k>0$, touch the $x$-axis at $(1,0)$. If the line $x+y=0$ intersects the circle $C$ at $P$ and $Q$ such that the length of the chord $PQ$ is 2, then the value of $h+k+r$ is equal to ______.

Let a circle C touch the lines $L_1:4x-3y+K_1=0$ and $L_2:4x-3y+K_2=0$, $K_1,K_2\in\mathbb R$. If a line passing through the centre of the circle C intersects $L_1$ at $(-1,2)$ and $L_2$ at $(3,-6)$, then the equation of the circle C is :

If $y=m_1x+c_1$ and $y=m_2x+c_2$, $m_1\ne m_2$ are two common tangents of circle $x^2+y^2=2$ and parabola $y^2=x$, then the value of $8|m_1m_2|$ is equal to :

Let the abscissae of the two points P and Q be the roots of $2x^2-rx+p=0$ and the ordinates of P and Q be the roots of $x^2-sx-q=0$. If the equation of the circle described on PQ as diameter is $2\left(x^2+y^2\right)-11x-14y-22=0$, then $2r+s-2q+p$ is equal to ______.

A circle touches both the $y$-axis and the line $x+y=0$. Then the locus of its center is:

Let C be a circle passing through the points A$(2,-1)$ and B$(3,4)$. The line segment AB is not a diameter of C. If r is the radius of C and its centre lies on the circle $(x-5)^2+(y-1)^2=\dfrac{13}2$, then $r^2$ is equal to :

Let the common tangents to the curves $4\left(x^2+y^2\right)=9$ and $y^2=4x$ intersect at the point Q. Let an ellipse, centered at the origin O, has lengths of semi-minor and semi-major axes equal to OQ and 6, respectively. If $e$ and $l$ respectively denote the eccentricity and the length of the latus rectum of this ellipse, then $\dfrac l{e^2}$ is equal to ______.

If $m$ is the slope of a common tangent to the curves $\dfrac{x^2}{16}+\dfrac{y^2}9=1$ and $x^2+y^2=12$, then $12m^2$ is equal to :

Let $\dfrac{dy}{dx}=\dfrac{ax-by+a}{bx+cy+a}$, where $a,b,c$ are constants, represent a circle passing through the point $(2,5)$. Then the shortest distance of the point $(11,6)$ from this circle is :

A rectangle R with end points of one of its sides as $(1,2)$ and $(3,6)$ is inscribed in a circle. If the equation of a diameter of the circle is $2x-y+4=0$, then the area of R is ______.

A circle of radius 2 unit passes through the vertex and the focus of the parabola $y^2=2x$ and touches the parabola $y=\left(x-\dfrac14\right)^2+\alpha$, where $\alpha>0$. Then $(4\alpha-8)^2$ is equal to ______.

The set of values of $k$, for which the circle $C:4x^2+4y^2-12x+8y+k=0$ lies inside the fourth quadrant and the point $\left(1,-\dfrac13\right)$ lies on or inside the circle C, is

Let a circle C of radius 5 lie below the $x$-axis. The line $L_1:4x+3y+2=0$ passes through the centre P of the circle C and intersects the line $L_2:3x-4y-11=0$ at Q. The line $L_2$ touches C at the point Q. Then the distance of P from the line $5x-12y+51=0$ is ______.

If the tangents drawn at the points $O(0,0)$ and $P\left(1+\sqrt5,2\right)$ on the circle $x^2+y^2-2x-4y=0$ intersect at the point Q, then the area of the triangle OPQ is equal to

Let the lines $y+2x=\sqrt{11}+7\sqrt7$ and $2y+x=2\sqrt{11}+6\sqrt7$ be normal to a circle $C:(x-h)^2+(y-k)^2=r^2$. If the line $\sqrt{11}y-3x=\dfrac{5\sqrt{77}}3+11$ is tangent to the circle C, then the value of $(5h-8k)^2+5r^2$ is equal to ______.