The length of the chord intercepted by the circle $x^2+y^2-4x+6y-12=0$ on the line $4x+3y+1=0$ is

The minimum distance and maximum distance of the point $P(2,-7)$ from the circle $x^2+y^2-14x-10y-151=0$ are respectively . . . . . . units.

$A$ rectangle is inscribed in a circle with a diameter lying along the line $3y = x + 7$. If the two adjacent vertices of the rectangle are $(-8, 5)$ and $(6, 5)$,then the area of the rectangle (in $sq. units$) is

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

Let $x_0, y_0$ be fixed real numbers such that $x_0^2+y_0^2 > 1$. If $x, y$ are arbitrary real numbers such that $x^2+y^2 \leq 1$,then the minimum value of $(x-x_0)^2+(y-y_0)^2$ is

If two parallel chords of a circle,having diameter $4 \, \text{units}$,lie on the opposite sides of the centre and subtend angles $\cos^{-1}\left(\frac{1}{7}\right)$ and $\sec^{-1}(7)$ at the centre respectively,then the distance between these chords is: