If three complex numbers are in A.P. then they lie on a circle in the complex plane.

From the origin chords are drawn to the circle $${(x - 1)^2} + {y^2} = 1$$. The equation of the locus of the mid-points of these chords is.............

No tangent can be drawn from the point (5/2, 1) to the circumcircle of the triangle with vertices $$\left( {1,\sqrt 3 } \right)\,\,\left( {1, - \sqrt 3 } \right),\,\,\left( {3,\sqrt 3 } \right)$$.

If $${f_r}\left( x \right),{g_r}\left( x \right),{h_r}\left( x \right),r = 1,2,3$$ are polynomials in $$x$$ such that $${f_r}\left( a \right) = {g_r}\left( a \right) = {h_r}\left( a \right),r = 1,2,3$$
and $$F\left( x \right) = \left| {\matrix{ {{f_1}\left( x \right)} & {{f_2}\left( x \right)} & {{f_3}\left( x \right)} \cr {{g_1}\left( x \right)} & {{g_2}\left( x \right)} & {{g_3}\left( x \right)} \cr {{h_1}\left( x \right)} & {{h_2}\left( x \right)} & {{h_3}\left( x \right)} \cr } } \right|$$ then $$F'\left( x \right)$$ at $$x = a$$ is ...........