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

[Shamako]

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 o the slope of the tangent line. Assume y` > 0 and y > 0.

Hint: $\displaystyle \int\frac{\sqrt{a^{2} - u^{2}}}{u}du = \sqrt{a^{2}-u^{2}} - a*ln |\frac{a+\sqrt{a^{2}-u^{2}}}{ u} | + C$
(sorry, I don't know how to use the math writer yet)

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:
$\displaystyle y_{t} = \frac{df}{dx}x + C$
Normal line will have equation $\displaystyle y_{n} = \frac{-1}{df/dx}x + k$

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 how to do this! 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!

 

[Deleted member 4993]

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 o the slope of the tangent line. Assume y` > 0 and y > 0.

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

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 how to do this! 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!

Duplicate Post:

http://answers.yahoo.com/question/index?qid=20111008172259AAorq1r

 

[Shamako]

It doesn't say I'm not allowed to ask this question on more than one website.... The reason I've asked it here is that its very rare for questions that actually involve more work than just solving an equation to actually be answered on yahoo answers, and I urgently need help on this.