$y - x + 3 = 0$ is the equation of normal at $\left( {3 + \frac{3}{{\sqrt 2 }},\frac{3}{{\sqrt 2 }}} \right)$ to which of the following circles

The angle at which the circles $(x - 1)^2 + y^2 = 10$ and $x^2 + (y - 2)^2 = 5$ intersect is

In the figure, $A B C D$ is a unit square. A circle is drawn with centre $O$ on the extended line $C D$ and passing through $A$. If the diagonal $A C$ is tangent to the circle, then the area of the shaded region is

If the line $y = \sqrt 3 x + k$ touches the circle ${x^2} + {y^2} = 16$, then $k =$

If the area of the triangle formed by the positive $x-$axis, the normal and the tangent to the circle $(x-2)^{2}+(y-3)^{2}=25$ at the point $(5,7)$ is $A$ then $24 A$ is equal to ...... .

From the origin chords are drawn to the circle ${(x - 1)^2} + {y^2} = 1$. The equation of the locus of the middle points of these chords is