Find all functions in 1. quadrant which tangts form triangles

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SUMMARY

The discussion revolves around finding functions in the first quadrant whose tangents form triangles with a constant area, denoted as P. The tangent line's slope is defined by the derivative, represented as k = (y2 - y1) / (x2 - x1) and is equal to y'(x). The problem requires deriving a differential equation from the geometric properties of the tangents and solving it to identify the functions that satisfy the area condition.

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  • Understanding of differential equations
  • Knowledge of calculus, specifically derivatives and tangent lines
  • Familiarity with geometric properties of triangles
  • Ability to manipulate algebraic expressions
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  • Learn about the properties of tangent lines in calculus
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Homework Statement


Find all functions for which any tangent in first quadrant "forms" a triangle with constant surface P. (You can assume that y'<0)



Homework Equations





The Attempt at a Solution


Now, I know I should somehow get to differential equation and then solve it but, I haven't got a clue on how to start?

Tangent is a linear function with ##k=\frac{y_2-y_1}{x_2-x_1}## and also equal to ##k=y^{'}(x)##..

That is in fact everything I have so far. I've been staring and this problem for quite a while now. Please help.
 
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Take a point on the curve, y = f(x). Where does the tangent at that point intersect the axes?
 

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