The number of circles touching the line $y - x = 0$ and the $y$-axis is

If the circles ${x^2} + {y^2} = 4,{x^2} + {y^2} - 10x + \lambda = 0$ touch externally, then $\lambda $ is equal to 

If the circles ${x^2} + {y^2} + 2ax + cy + a = 0$ and ${x^2} + {y^2} - 3ax + dy - 1 = 0$ intersect in two distinct points $P$ and $Q$ then the line $5x + by - a = 0$ passes through $P$ and $Q$ for

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 $(x-5)^{2}+(y-1)^{2}=\frac{13}{2}$, then $r^{2}$ is equal to

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

Consider a family of circles which are passing through the point $(- 1, 1)$ and are tangent to $x-$ axis. If $(h, k)$ are the coordinate of the centre of the circles, then the set of values of $k$ is given by the interval