consider the curve $y=x^2$
what are the points on the curve that are the closest to the point $(1,0)$
using calculus I got the two points but what is the connection between normals and the closest distance to a curve from a point
is it that drawing normals from 1,0 and seeing where it meets the curve gives the points closest to it – although a normal from 1,0 does go through the origin even though the origin is not the closest to the curve
please explain these observations
Best Answer
You may construct a cirle with centre in $A$ and radius $R$. It has the following equation: $$(x-x_a)^2 + (y-y_a)^2 = R^2$$ The lowest $R$ for which this circle intersects $f(x)$ is the solution of your problem. So you have to substitute $y$ in the equation above with expression for $f(x)$ and solve resulting equation for $x$. Then you have to find the lowest $R > 0$ for which exists at least one solution. This value for $R$ will be the answer.