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
$2\sqrt 3$
$\sqrt 3$
$1/\sqrt 3$
$1$
If line $ax + by = 0$ touches ${x^2} + {y^2} + 2x + 4y = 0$ and is a normal to the circle ${x^2} + {y^2} - 4x + 2y - 3 = 0$, then value of $(a,b)$ will be
Tangents are drawn from $(4, 4) $ to the circle $x^2 + y^2 - 2x - 2y - 7 = 0$ to meet the circle at $A$ and $B$. The length of the chord $AB $ is
Two tangents drawn from the origin to the circle ${x^2} + {y^2} + 2gx + 2fy + c = 0$ will be perpendicular to each other, if
The point at which the normal to the circle ${x^2} + {y^2} + 4x + 6y - 39 = 0$ at the point $(2, 3)$ will meet the circle again, is
From the origin chords are drawn to the circle ${(x - 1)^2} + {y^2} = 1$. The equation of the locus of the middle points of these chords is