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

The equations of the tangents drawn from the origin to the circle ${x^2} + {y^2} - 2rx - 2hy + {h^2} = 0$ are

Let the tangent to the circle $x^{2}+y^{2}=25$ at the point $R (3,4)$ meet $x$ -axis and $y$ -axis at point $P$ and $Q$, respectively. If $r$ is the radius of the circle passing through the origin $O$ and having centre at the incentre of the triangle $OPQ ,$ then $r ^{2}$ is equal to

A pair of tangents are drawn to a unit circle with centre at the origin and these tangents intersect at A enclosing an angle of $60^o$. The area enclosed by these tangents and the arc of the circle is

The equation of the tangent at the point $\left( {\frac{{a{b^2}}}{{{a^2} + {b^2}}},\frac{{{a^2}b}}{{{a^2} + {b^2}}}} \right)$ of the circle ${x^2} + {y^2} = \frac{{{a^2}{b^2}}}{{{a^2} + {b^2}}} $ is

Length of the tangent from $({x_1},{y_1})$ to the circle ${x^2} + {y^2} + 2gx + 2fy + c = 0$ is