If the circles $$x^2+y^2=9$$ and $$x^2+y^2+2 \alpha x+2 y+1=0$$ touch each other internally, then the value of $$\alpha^3$$ is

The sides of a rectangle are given by the equations $$x=-2, x=4, y=-2$$ and $$y=5$$

Then the equation of the circle, whose centre is the point of intersection of the diagonals, lying within the rectangle and touching only two opposite sides, is

Two tangents to the circle $$x^2+y^2=4$$ at the points $$\mathrm{A}$$ and $$\mathrm{B}$$ meet at the point $$\mathrm{P}(-4,0)$$. Then the area of the quadrilateral $$\mathrm{PAOB}, \mathrm{O}$$ being the origin, is

If the lines $$3 x-4 y-7=0$$ and $$2 x-3 y-5=0$$ pass through diameters of a circle of area $$49 \pi$$ square units, then the equation of the circle is