The locus of the mid-point of the line segment joining the focus to a moving point on the parabola $${y^2} = 4ax$$ is another parabola with directrix

The equation of the common tangent to the curves $${y^2} = 8x$$ and $$xy = - 1$$ is

Which of the following pieces of data does NOT uniquely determine an acute-angled triangle $$ABC$$ ($$R$$ being the radius of the circumcircle)?

The length of a longest interval in which the function $$3\,\sin x - 4{\sin ^3}x$$ is increasing, is