# Perpendicular to a curve

• pixel01

#### pixel01

Can anyone remind me about this. In a plane, there a given curve y=f(x). Now, from a given point on the plane, i can draw a line which is perpendicular to the curve (can be zero, one,two, three ..lines). I can't remember what the equation describing this line(s) is.
Thanks.

That's because there is not simple equation for it!

Here's how I attempt to find the equation. The perpendicular from a point, (a, b) to the curve (x, f(x)) gives the (locally) shortest distance to the curve. The distance (squared) from (a, b) to (x, f(x)) is (x-a)2+ (f(x)- b)2. Differentiating that, 2(x-a)+ 2(f(x)- b)f'(x)= 0 for the closest point. x must satisfy the equation x- a+ (f(x)- b)f'(x)= 0. That can be solved for specific f but I see no way to get a general equation.

Assuming you've got a nice function, the first derivative gives the tangent to the curve. The perpendicular to the curve at a given point is the same as the perpendicular to the tangent at that point. Specifically, the tangent at the point x is a line through x with slope f'(x). The perpendicular through the same point has slope -1/f'(x).