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

If the centre of a circle is $(-6, 8)$ and it passes through the origin,then the equation of its tangent at the origin is

The area of the triangle formed by the positive $X$-axis, the tangent and the normal to the curve $x^2+y^2=16a^2$ at the point $(2\sqrt{2}a, 2\sqrt{2}a)$ is

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

The angle between the tangents drawn from the point $P(k, 6k)$ to the circle $x^2+y^2+6x-6y+2=0$ is $2 \operatorname{Tan}^{-1}\left(\frac{4}{3}\right)$. If the coordinates of $P$ are integers,then $k=$

The circle $x^2+y^2-4x-8y+16=0$ rolls along the tangent drawn to it at $(2+\sqrt{3}, 3)$ by $2$ units. The equation of the circle in the new position is