3680.21 – Grasshopper Jump

A grasshopper jumps from its position at $(x,y)$, over a proud blade of grass at point $(a,b)$, to the point that is diametrically opposite to $(x,y)$. In other words, the three points are collinear, and the grasshopper’s new position is just as far from $(a,b)$ as it was before. What are the coordinates of this new point, in terms of $a$, $b$, $x$, and $y$?

Solution

The landing point is $(2a – x, 2b – y)$.