If the chord $x+y=1$ of the circle $x^2+y^2=a^2$ subtends a right angle at the origin,then $a=$

Two circles of radii $4 \text{ cm}$ and $1 \text{ cm}$ touch each other externally and $\theta$ is the angle contained by their direct common tangents. Then $\sin \theta =$

When do the circles represented by the equations $x^{2} + y^{2} + c^{2} = 2ax$ and $x^{2} + y^{2} + c^{2} = 2by$ touch each other externally?

Let the centre of the circle $S=0$ lie on the line $x+y-5=0$ and also lie in the first quadrant. If this circle touches both the lines $x-2=0$ and $y-5=0$,then the area of the circle is

From a point $P$ on the line $4x - 3y = 6$,two tangents are drawn to the circle $x^2 + y^2 - 6x - 4y + 4 = 0$. If the angle between these tangents is $\tan^{-1}\left(\frac{24}{7}\right)$,then $P$ can be:

Lines are drawn from a point $P(-1, 3)$ to a circle $x^2 + y^2 - 2x + 4y - 8 = 0$. If the line meets the circle at two points $A$ and $B$,then the minimum value of $PA + PB$ is: