If the line $y = \sqrt{3}x + k$ touches the circle $x^2 + y^2 = 16$,then $k =$

If the line $x+y=0$ touches the curve $ax^2 = 2y^2 - b$ at $(1, -1)$,then the values of $a$ and $b$ are respectively:

The equations of the two tangents from $(-5, -4)$ to the circle $x^{2}+y^{2}+4x+6y+8=0$ are

Let the tangents drawn to the circle $x^2 + y^2 = 16$ from the point $P(0, h)$ meet the $x-$axis at points $A$ and $B$. If the area of $\Delta APB$ is minimum, then $h$ is equal to

The gradient of the tangent line at the point $(a \cos \alpha, a \sin \alpha)$ to the circle $x^2 + y^2 = a^2$ is

The equation of the common tangent touching the circle $(x-3)^2+y^2=9$ and the parabola $y^2=4x$ above the $X$-axis is