The value of $c$ for which the set,$\{(x, y) | x^2 + y^2 + 2x \le 1 \} \cap \{(x, y) | x - y + c \ge 0\}$ contains only one point in common is :

If the lines $2x + y + 12 = 0$ and $kx - 3y - 10 = 0$ are conjugate with respect to the circle $x^2 + y^2 - 4x + 3y - 1 = 0$,then $k =$

If two chords,each bisected by the $x$-axis,can be drawn to the circle $2(x^2 + y^2) - 2ax - by = 0$ $(a \ne 0, b \ne 0)$ from the point $P(a, b/2)$,then:

If the length of the tangents drawn 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 $k =$

The equation of the circle symmetric to the circle $x^2 + y^2 - 2x - 4y + 4 = 0$ about the line $x - y = 3$ is

Given: $A$ circle $2x^2 + 2y^2 = 5$ and a parabola $y^2 = 4\sqrt{5}x$.
Statement-$1$: An equation of a common tangent to these curves is $y = x + \sqrt{5}$.
Statement-$2$: If the line $y = mx + \frac{\sqrt{5}}{m} (m \neq 0)$ is their common tangent,then $m$ satisfies $m^4 - 3m^2 + 2 = 0$.