Given the curve the slope of the tangent at is and the slope of the normal at is .
Now write the equations of the two lines.
Find the point of their intersection.
Find the derivative of
Slope at is a simple plug and chug of the x-coordinate, so
Slope at is just , so the line normal to this point is perpendicular. Perpendicular slopes are opposite signed reciprocals, so that's how Plato got
set up three simultaneous equations. since (1,2) and (2,1) are on the curve, we have:
f(1) = 2 and f(2) = 1. so form two equations from that.
but there are three unknowns, we need at least three equations. so use the information from the derivative. we are also told that f ' (2) = 0, that's your third equation