The angle between the tangents from $(\alpha, \beta)$ to the circle $x^2 + y^2 = a^2$ is

The area (in sq units) of the triangle formed by the tangent,normal at $(1, \sqrt{3})$ to the circle $x^2+y^2=4$ and the $X$-axis,is

The angle of intersection between the curves $y=[|\sin x|+|\cos x|]$ and $x^{2}+y^{2}=10,$ where $[x]$ denotes the greatest integer $\leq x,$ is

The point on the curve $x^2+y^2=a^2, y \geq 0$,at which the tangent is parallel to the $x$-axis is

Find the points on the curve $x^{2}+y^{2}-2x-3=0$ at which the tangents are parallel to the $x$-axis.

Let $ABCD$ be a square of side length $1$. $A$ circle $C_{1}$ centered at $A$ with unit radius is drawn. Another circle $C_{2}$ which touches $C_{1}$ and is tangent to the lines $AD$ and $AB$ is also drawn. Let a tangent line from the point $C$ to the circle $C_{2}$ meet the side $AB$ at $E$. If the length of $EB$ is $\alpha+\sqrt{3} \beta,$ where $\alpha, \beta$ are integers,then $\alpha+\beta$ is equal to.........