The equation of the circle having its centre on the line $x + 2y – 3 = 0$ and passing through the points of intersection of the circles ${x^2} + {y^2} – 2x – 4y + 1 = 0$ and ${x^2} + {y^2} – 4x – 2y + 4 = 0$, is

Let $r_{1}$ and $r_{2}$ be the radii of the largest and smallest circles, respectively, which pass through the point $(-4,1)$ and having their centres on the circumference of the circle $x^{2}+y^{2}+2 x+4 y-4= 0.$ If $\frac{r_{1}}{r_{2}}=a+b \sqrt{2}$, then $a+b$ is equal to:

The equation of a circle passing through origin and co-axial to circles ${x^2} + {y^2} = {a^2}$ and ${x^2} + {y^2} + 2ax = 2{a^2},$ is

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:

If the circle ${x^2} + {y^2} + 6x – 2y + k = 0$ bisects the circumference of the circle ${x^2} + {y^2} + 2x – 6y – 15 = 0,$ then $k =$