The length of the tangent from the point $(5, 1)$ to the circle $x^2 + y^2 + 6x - 4y - 3 = 0$ is:

$A$ 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

$A$ circle $C$ touches the $X$-axis and makes an intercept of length $2$ units on the $Y$-axis. If the centre of this circle lies on the line $y=x+1$,then which of the following is a circle passing through the centre of the circle $C$?

The number of circles that touch all the three lines $x+y-1=0$,$x-y-1=0$,and $y+1=0$ is

The line $4x - 3y + 2 = 0$ intersects the circle $x^2 + y^2 - 2x + 6y + c = 0$ at two points $A$ and $B$,and the length of the chord $AB = 8$. If $(1, k)$ is a point on the given circle and $k > 0$,then $k =$

Find the equation of the chord of the circle $x^2 + y^2 = 16$ which is bisected at the point $(2, -1)$.