If the circle $(x - h)^2 + (y - k)^2 = r^2$ touches the curve $y = x^2 + 1$ at the point $(1, 2)$,then the possible locations of the points $(h, k)$ are given by

If the angle between the circles $x^2+y^2-2x-4y+c=0$ and $x^2+y^2-4x-2y+4=0$ is $60^{\circ}$,then $c=$

If the equation of the circle whose radius is $\sqrt{10}$ and which touches the circle $x^2+y^2+2x+8y-23=0$ externally at the point $(1,2)$ is $x^2+y^2+ax+by+c=0$,then $|a+b+c|=$

Let the centre of the circle be in the first quadrant and lie on the line $2x - y = 4$. Let the area of an equilateral triangle inscribed in the circle be $27\sqrt{3}$. Then the square of the length of the chord of the circle on the line $x = 1$ is . . . . . .

If the length of the chord of the circle $x^{2}+y^{2}=r^{2}$ $(r>0)$ along the line $y-2x=3$ is $r$,then $r^{2}$ is equal to

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 $|\alpha| =$