The perimeter of a certain sector of a circle is equal to the length of the arc of the semicircle. Then,the angle at the centre of the sector in radians is

Let $O$ be the centre of the circle $x^2 + y^2 = r^2$,where $r > \frac{\sqrt{5}}{2}$. Suppose $PQ$ is a chord of this circle and the equation of the line passing through $P$ and $Q$ is $2x + 4y = 5$. If the centre of the circumcircle of the triangle $OPQ$ lies on the line $x + 2y = 4$,then the value of $r$ is:

If the circles $x^2 + y^2 = a^2$ and $x^2 + y^2 - 2gx + g^2 - b^2 = 0$ touch each other externally,then:

The least and the greatest distances of the point $(10, 7)$ from the circle $x^{2} + y^{2} - 4x - 2y - 20 = 0$ are

The curve $xy = c, (c > 0)$,and the circle $x^2 + y^2 = 1$ touch at two points. Then the distance between the points of contact is

If the area of the circum-circle of the triangle formed by the line $2x + 5y + \alpha = 0$ and the positive coordinate axes is $\frac{29\pi}{4}$ sq. units,then $|\alpha| =$