Find curve with tangent and normal lines that create a triangle with given area

[Shamako]

Homework Statement

Find the implicit equation of the curve that goes through the point (3, 1) and whose tangent and normal lines always form with the x-axis a triangle whose area is equal to the slope of the tangent line. Assume y` > 0 and y > 0.

Homework Equations

Hint: ∫( √(a^2 - u^2) / u du = √(a^2-u^2) - a*ln | [a+√(a^2-u^2)] / u | + C
(sorry, I don't know how to use the math writer yet)

The Attempt at a Solution

This is a question from an introductory differential equations class. I have absolutely no idea how to do this! I haven't really gotten anywhere yet. This is what I've done:

let f(x) denote the curve we're looking for. Then the tangent line will have equation:
y_t = df/dx * x + C
Normal line will have equation y_n = -1/(df/dx) * x + k

Together they will form a triangle with area = df/dx, at any point on f(x). I wanted to find an expression for area in terms of df/dx, simplify it, and solve the resulting differential equation, but I can't figure out a DE for the area! I'm getting very frustrated, as we've never been shown a question like this in lecture, and I can't find any examples in my textbook.

Help would be very much appreciated!

 

[teyber]

Moore's 201 class at uvic? same boat...

the only useful thing I've written down is dT/dx = (1/2)(T)(N)

its dT/dx because its equal to the slope of the tangent line

Please post back with any progress you make and ill do the same