A spacecraft is due to make a close pass of an object in space and scientists would like to investigate the object more carefully using a telescope on-board the spacecraft. For simplicity, let us reduce the problem to two dimensions and assume that the position of the space craft is stationary in $(0,0)$ and the shape of the object is a disk and its boundary has the equation

    $$x^2 + y^2 − 10x − 8y + 40 = 0$$

    Find the maximum value of $\rm tan \phi$ where $\phi$ is the elevation angle of the telescope with respect to the “horizontal” direction (x-axis) during investigation from one edge to the other edge.

Source: IOAA