If $z_1$ is a point on $z\bar{z} = 1$ and $z_2$ is another point on $(4 - 3i)z + (4 + 3i)\bar{z} - 15 = 0$,then $|z_1 - z_2|_{min}$ is (where $i = \sqrt{-1}$)

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 $k =$

If a circle passes through the points of intersection of the lines $\lambda x - y + 1 = 0$ and $x - 2y + 3 = 0$ with the coordinate axes,then the value of $\lambda$ is:

Tangent $L_1 \equiv 3x - 4y - 8 = 0$ and the chord $L_2 \equiv x + y - 1 = 0$ are at a distance of $2$ and $\sqrt{2}$ units respectively from the centre of a circle $S$. $(h, k)$ is the centre of $S$ such that $h^2 + k^2 = 13$. If the midpoint of the chord $L_2 = 0$ is $(\alpha, \beta)$ and the radius of the circle is $r$,then $\alpha + \beta + r =$

The area of the circle passing through the points $(5, 2), (5, -2),$ and $(1, 2)$ is (in $\pi$)

The equation of the circle having the lines $x^2 + 2xy + 3x + 6y = 0$ as its normals and having a size just sufficient to contain the circle $x(x - 4) + y(y - 3) = 0$ is