The area of the circle having its centre at $(3,4)$ and touching the line $5x+12y-11=0$ is

If the chord $y = mx + 1$ of the circle ${x^2} + {y^2} = 1$ subtends an angle of measure ${45^\circ}$ at the major segment of the circle,then the value of $m$ is:

The least distance of the point $(10, 7)$ from the circle $x^2 + y^2 - 4x - 2y - 20 = 0$ is

If $(a \cos \theta_i, a \sin \theta_i)$ for $i = 1, 2, 3$ represent the vertices of an equilateral triangle inscribed in a circle $x^2 + y^2 = a^2$,then:

Let $AB$ be a line segment of length $2$. Construct a semicircle $S$ with $AB$ as diameter. Let $C$ be the mid-point of the arc $AB$. Construct another semicircle $T$ external to the $\triangle ABC$ with chord $AC$ as diameter. The area of the region inside the semicircle $T$ but outside $S$ is

The acute angle between the curves $x^2+y^2=x+y$ and $x^2+y^2=2y$ is