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.
You are probably wondering how Plato got those slopes.
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
let f(x) = ax^2 + bx + c.
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