Family of 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 _____.

If the circles $x^2+y^2+6x+8y+16=0$ and $x^2+y^2+2\left(3-\sqrt3\right)x+2\left(4-\sqrt6\right)y=k+6\sqrt3+8\sqrt6$, $k>0$, touch internally at the point $P(\alpha,\beta)$, then $\left(\alpha+\sqrt3\right)^2+\left(\beta+\sqrt6\right)^2$ is equal to ______.

The differential equation of the family of circles passing through the points (0, 2) and (0, −2) is :

Let $O$ be the centre of the circle $x^2 + y^2 = r^2$, where $r > \dfrac{\sqrt5}{2}$. Suppose $PQ$ is a chord of this circle and the equation of the line passing through $P$ and $Q$ is $2x + 4y = 5$. If the centre of the circumcircle of the triangle $OPQ$ lies on the line $x + 2y = 4$, then the value of $r$ is ______

Which of the following is the only CORRECT combination?

Which of the following is the only INCORRECT combination?

The centres of those circles which touch the circle, $x^2+y^2-8x-8y-4=0$, externally and also touch the $x$-axis, lie on :

Consider the family of all circles whose centers lie on the straight line $y=x$. If this family of circles is represented by the differential equation $Py''+Qy'+1=0$, where $P$, $Q$ are functions of $x$, $y$ and $y'$ (here $y'=\dfrac{dy}{dx}$, $y''=\dfrac{d^2y}{dx^2}$), then which of the following statements is (are) true?

A circle $S$ passes through the point $(0,1)$ and is orthogonal to the circles $(x-1)^2+y^2=16$ and $x^2+y^2=1$. Then

The set of all real values of $\lambda$ for which exactly two common tangents can be drawn to the circles $x^2+y^2-4x-4y+6=0$ and $x^2+y^2-10x-10y+\lambda=0$ is the interval :

The equation of the circle described on the chord $3x+y+5=0$ of the circle $x^2+y^2=16$ as diameter is :