The area of the triangle formed by the tangent at $(3, 4)$ to the circle ${x^2} + {y^2} = 25$ and the coordinate axes is

If $\Delta$ is the area of the triangle formed by the positive $x$-axis and the normal and tangent to the circle $x^2+y^2=4$ at $(1, \sqrt{3})$,then $\Delta$ is equal to

Find the equations of the tangents drawn to the circle $x^2+y^2=50$ at the points where the line $x+7=0$ meets it.

Tangents drawn from the origin to the circle $x^2 + y^2 - 2ax - 2by + b^2 = 0$ are perpendicular to each other,if

If the tangent at the point $P$ on the circle $x^2+y^2+6x+6y=2$ meets the straight line $5x-2y+6=0$ at a point $Q$ on the $Y$-axis,then the length of $PQ$ is

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.........