Given three collinear points $A(3,1)$,$B(7,-1)$,and $C(5,0)$. The length of a tangent drawn from $A$ to any circle that passes through $B$ and $C$ is ....... units.

Let $C$ be a circle having centre in the first quadrant and touching the $x$-axis at a distance of $3$ units from the origin. If the circle $C$ has an intercept of length $6\sqrt{3}$ on the $y$-axis, then the length of the chord of the circle on the line $x - y = 3$ is:

The angle of intersection of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ and the circle $x^2 + y^2 = ab$ is

The lengths of the tangents from the point $(1,2)$ to the circles $x^2+y^2+x+y-4=0$ and $3x^2+3y^2-x-y-k=0$ are in the ratio $4:3$. Then the value of $k$ is:

Choose the incorrect statement about the two circles whose equations are given below:
$x^{2}+y^{2}-10x-10y+41=0$ and $x^{2}+y^{2}-16x-10y+80=0$

If $P(\frac{\pi}{3})$ and $Q(\frac{2\pi}{3})$ represent two points on the circle $x^2+y^2-4x+6y-12=0$ in parametric form,then the length of the chord $PQ$ is