An equilateral triangle $OAB$ is inscribed in the parabola $y^{2}=4x$ with the vertex $O$ at the origin. Then the minimum distance of the circle having $AB$ as a diameter from the origin is:

If a line along a chord of the circle $4x^{2}+4y^{2}+120x+675=0$ passes through the point $(-30, 0)$ and is tangent to the parabola $y^{2}=30x$,then the length of this chord is:

$A$ pair of tangents are drawn to a unit circle with centre at the origin and these tangents intersect at a point $A$ enclosing an angle of $60^o$. The area enclosed by these tangents and the arc of the circle is

$A(x_1, y_1)$ is the internal centre of similitude and $B(x_2, y_2)$ is the external centre of similitude of two circles $C_1$ and $C_2$ whose centres are $P(\alpha, \beta)$ and $Q(\gamma, \delta)$ respectively. If $PA=3, AB=5, QB=2$,then the ratio of the radii of the two circles is:

If the point $(1, 4)$ lies inside the circle $x^2 + y^2 - 6x - 10y + p = 0$ and the circle does not touch or intersect the coordinate axes,then the set of all possible values of $p$ is the interval

The equation of a common tangent to the circle $x^2+y^2=16$ and the ellipse $\frac{x^2}{49}+\frac{y^2}{4}=1$ is